Francis John George Gibbons

A project report submitted to the Faculty o£ University of the Wltwatersrand, Johannesburg, fulfilment of the requirements for the degree Science in Engineering

Engineering, in partial

of Master of

J o h a n n e s b u r g , 1986

I declare that this project report is my owm, unaided » is being submitted foi the Degree of Master ef Sc Engineering in the University of the tfltwatJohannesburg, it has not been submitted before for aii or examination in any other Vnivoraity,

'/T " .er. U '■

k flood routing model 1 cions for unsteady om project report. The i

reach as well aa the mi The model does r the physical characteristics o£ the river as defined for a

'water computation being given, and can therefore be ungauged watercourses with more confidence than the

flood routing models. The model ie based on i difference scheme, being Verwey’s variant of

the Preissman Scheme. It uses the double sweep algorithm for the solution of the difference equations and is therefore restricted to subcritical flows.

aed on the complete de St. Venant equa- dimensional flow is described in this del enables the user to determine the the flood peak through a single river

iched by the flood, i of routing constants,

ACKNOWLEDGEMENTS

I wish to express my gratitude to the following.

Professor David Stephenson, my supervisor, for . encouragement received through the duration of t

Scott & de Waal Incorporated, for the use of Facilities.

Val Duminy for her patience in typing the numero

My wife, Georgia, for tracing the figures and

all the help and his project.

their Computer

us drafts,

providing moral

DECLARATIONABSTRACTACKNOWLEDGEMENTSCONTENTSLIST OF FIGURESLIST OF TABLESLIST OF SYMBOLS

1. INTRODUCTION AND LITERATURE SURVEY

1.1 Introduction1.2 A Review of Approximate Flood Routing Methods1.2.1 Kinematic models1.2.2 Approximate dynamic and diffusion analogy models1.2.3 Applicability of the approximate

flood routing methods1.3 Complete Dynamic Models1.3.1 The method of characteristics1.3.2 Finite difference methods1.3.3 Explicit finite difference schemes1.3.4 Implicit finite difference methods1.3.5 Comparison of explicit and implicit schemes 1.4 Selection of Flood Routing Method

2. DESCRIPTION OF THE FINITE DIFFERENCE SCHEME OF VERWEY

2.1 The de St. Tenant Equations for One Dimensional Flow in Natural Channels

2.2 Discretization of the Flow Equations2.3 Double Sweep Algorithm2.4 Initial and Boundary Conditions

3.1.4

3.23.3

DESCRIPTION OF COMPUTER PROGRAM

Data Input Cross-section data Inflow hydrograph data Downstream boundary condition Initial conditionsSelection of the weighting coefficients pad time step Ma? - Coapiitation Presentation of Results

APPLICATION OF THE MODEL

IntroductionThe Svartvlei EstuaryThe Rietspruit Watercourse

SUMMARY AMD CONCLUSION

SummaryConclusion

APPSTOIX A USER INSTRUCTIONSFLOW_KOD 92BACKDATA 103

APPENDIX B PROGRAM LISTINGS 109FLOWJiOD .110BACKDATA 12?

APPENDIX C SWARTVLEI ESTUARY 133CROSS-SECTION DATA 134RESULTS OF TIDAL RUN 143

,APPENDIX D RIETSPRUIT WATERCOURSE CROSS-SECTION DATA 147

REFERENCES

I

LIST OF FIGURES

id rating curve grid £or kinematic models

characteristic method cs on a variable grid

discretization schemes (b) Leap-Frog

explicit discretization scheme ability and instability £or

id for box schemes

m of Che double sweep algorithm

. for the main computation

lei estuarydischarge hydrographs for 360 hour run

ie for Swartvlei estuary of measured and predicted water levels of measured and predicted flows catchmentographs used in model runs igraph with Hydrograph No.2

- 0,5hydrograph with Hydrograph No.2 1,0hydrograph with Hydrograph No.2

- 1,0>w hydrograph with Hydrograph No.3

LIST OF TABLES

Table Page

3.1 Sample listing of croas-section data file 55

4.1 Minimum values of 8 - Hydrograph No.1 83

4.2 Minimum values of 6 - Hydrograph No.2 83

4.3 Minimum values of 6 - Hydrograph No.3 83

‘J

LIST OF SYMBOLS

A - croos-aectional area (L^)A1, A2 - coetficlenV.a used in discretized flour equations

Ba - channel width (L)Bl, B2 - coefficients used in discretized flow equations

c - propagation veloc'-cy (L/T)Cl, C2 - coefficients used in discretized flow equations

D - diffusion coefficientDl, D2 - coefficients used in discretized flow equations

El, E2 - coefficients used in discretized flow equations

F - coefficient used in double sweep algorithm?£ - bed friction force acting on control volume (?)Fg ~ gravitational force acting on control volume (F) Fp - pressure force acting on control volume (F)F- - Froude number

- acceleration due to gravity (L/T^)- coefficient used in double sweep algorithm

- water depth (L)- coefficient used in double sweep algorithm

- computational point inde?- coefficient used in double sweep algorithm

• coefficient used in double sweep algorithm

- conveyance (L^/T)- travel time parameter

- routing coefficient

- Manning coefficient- time step index

- wetted perimeter (L)

- volumetric water discharge (l V t )- uniform flow rate for given depth of flow (tVl)- continuous lateral inflow per unit length (L^/T)

- hydraulic radius (L)

energy line slope in x-direction (friction slope)- bed slope in x-direction

wave period (1)- time (T)- time between two computational intervals (T)

- water velocity in the x-di- uniform flow velocity (L/T)- component of lateral inflow velocity in the

x-direction

- longitudinal space co-ordinate (I.)- d • i.nce between two computational poins in

x . section (L)

- vertical space co-ordinate above datum (L)- water surface elevation (L)- bed elevation (L)- weir crest level (L)- routing coeffient- non-uniform kinetic energy i-ivi-cfion coefficient

- non-uniform momentum correcii. ^efficient

- distance of cross-section c<.nt*\>id from water surface (L)

• weighting coefficient in finite difference approxi­mations of functions and their space derivatives

• water mass density (M/L^)• width of cross-section (L)• combined resistance and bed slope terms• weighting coefficient in finite difference approxi­mations of functions and their time derivatives

■ dimension less wave period• weir discharge coefficient■ coefficient used in double sweep algorithm

1. INTRODUCTION AND LITERATORS REVIEW

1.1 introduction

The ravages of flooding threugh the ages have cost man dearly in terms of life and property. Villages and portions of towns situatod on the banks of rivers have been washed away. Fertile lands on river flood plains have been destroyed by the deposition of silt by the sediment-laden flood-waters. Disruptions to communications and transport links end essential services have also resulted from severe flooding.

Knowledge of past flood events along a river provides the in­centive for taking adequate flood protection measures and the curtailment of development on unprotected flood plains. Where cities have existed for centuries on the banks of rivers this knowledge has accumulated so as to provide a sound statistical, basis for flood prediction. In a developing country such as South Africa flood records covering periods longer than sixty years at a particular river gauging station are rare. Little historical knowledge is therefore available for many rivers and watercourses regarding the probability distribution of the maximum flood levels reached during a flood event.

As development in a watershed occurs, the run-off characteristics of the watershed change. In general the same storm event will result in a run-off hydrograph from a developed watershed with a higher peak and shorter time to peak than that from the watershed in its undeveloped state. Furthermore, with the development in the catchment there is increasing utilization of flood plains of the watercourses for industrial, residential and agricultural purposes as the value of the land increases. Without judicious flood plain management this would result in drastic increases in damage as a result of flooding.

Statutory limitations on township development in flood plains do exist. For any watercourse traversing s proposed township development in South Africa it is a requirement of the Water Act (No. 54 of 1956 as amended) that the twenty year floodline for

I

\

the watercourse be determined. In some provinces additional restrictions are applied; in Che Transvaal the 50 year floodline is required by the Directr-r of Local Government. Although these floodlines do not serve as limits of the flood plain in which no development whatsoever is permitted, special flood protection measures are required where the developer wishes to extend into this area.

Through the use of the urban drainage models such as WITWAT and ILLUDAS, better descriptions of the run-off process within developed catchments are being obtained, facilitating the design of the stormwater systems for these areas. However, the point at which the stormwater systems discharge into the natural water­course is generally the limit of applicability of these models.

Once the characteristics of the flood hydr "’graph entering a particular reach of a natural watercourse have been estimated, the determination of the maximum levels reached by the flood is ahydraulic problem rather than a hydrological one. In some cases it is sufficient to carry out a backwater computation using the peak flow obtained from the hydrological analysis to determine these levels, thus assuming steady flow conditions. The effect of a volume of run-off entering a river channel in a relatively short period is to create a flood wave which progresses down the river. This unsteady flow condition can give rise to two effects ignored by the steady flow assumption. Firstly, if at a,single point in the river reach observations of discharge and stage are madeduring the passage of the flood wave, the stage discharge or rating curve formed from s' plot of the results will generally exhibit a looped form. Thus for any particular water level two flows are measured, the higher during the period that the flood is rising, the lower when the flood is falling. This is explained bythe fact that the water surface slope is steeper before the peakof the floodwave than it is after the peak. The result of this is that often the peak flow does not coincide with the maximum flood level, the latter occurring some, time after the peak flow.

The second effect ignored by the steady state assumption is that the shape of the flood wave changes as it progresses downstream.

The degree of change depends on the shape of the inflow hydrograph and the properties of the river channel. The most important of these properties are the amount of storage in the reach relative to the volume of the flood and Che travel time through the reach relative to the duration of the flood. In the case of water­courses with wide flood plains which drain developed catchments characterized by sharp-peaked run-off hydrographa attenuation tanbe significant.

The process by which the characteristics such as the speed, shape and height of a flood wave in a channel a:;e traced as these vary with time ia termed flood routing. The use of flood routing tech­niques to determine the maximum flood levels or the peak flow along a reach can lead to narrower floodline zoning along a water­course and more economical design of bridges and structures within the flood plain than would be obtained using simple time lag rout­ing and assuming steady flow conditions. Indeed, in the condi­tions described above where attentuation is significant, the savings will also be significant.

A number flood routing techniques of varying complexity have been developed. At the one end of the scale are the simplified hydro- logical routing techniques with which the discharge hydrographa at intervals along the reach can be obtained. The Muskingum method is a well-known and commonly used example of this type of routing technique. At the other end of the scale are the so-called complete dynamic models based on the complete hydrodynamic equations for unsteady flow and with which the discharge and water •level can be traced at all points along the reach for the duration of the flood. Obviously the data requirements for these latter techniques are greater than for the simpler techniques. However, correct reproduction of flood wave motion using the simplified techniques ia dependent upon having historical flood data for the reach in question with which to determine the routing constants required. This pertains especially to those simplified techniques based on the kinematic wave equation, discussed futher in Section 1.2.1, where the attenuation obtained in the solution arises solely out of the numerical damping inherent in the technique.

While calibration of the complete dynamic models is also not possi­ble without historical flood data, in the absence of flood records the complete dynamic models will provide a better simulation of Che flood wave motion than the simplified techniques. Whereas the routing constants for the simplified techniques have to be "guesstimated" in these situations, the complete dynamic models utilise actual cross-section data in the routing procedure, although the channel and flood plain roughness coefficients have to be estimated, and a better description of the hydraulic charac­teristics of the reach is obtained.

The second disadvantage with the use of the simplified techniques is that in most cases a backwater computation is still required to determine the maximum flood levels along the reach. To carry out this backwater computation cross-section data and estimates of the channel and flood plain roughness coefficients are required. The data requirements for carrying out a simplified flood routing pro­cedure and backwater computation are thus almost the same as those required for a run of a complete dynamic model.

A third disadvantage with the simplified models is that their range of applicability is limited in comparison with the complete dynamic models. The kinematic models are limited to relatively steep watercourses where backwater effects are negligible and to mildly sloped hydrographs. The simplified models based on the diffusion equation, which are discussed further in Section 1.2.2, have a far greater range of applicability than -the kinematic models and are suitable for most flood routing exercises. However, the diffusion models do not offer any real advantage over the complete dynamic models in terms of computational speed or simplicity of input data when applied to natural channels of varying cross-section.

The objective of this project has thus been to develop a complete dynamic flood routing model for general application to single channel reaches which utilises the actual cross-seetiona of the watercourse in the routing procedure and frcsi which maximum flood levels along the reach can be obtained. The model is based on an implicit finite difference scheme, being Verwey's variant of the

Preissmann scheme. As is discussed in Section 1.3.5, implicit finite difference schemes are far more suitable than explicit schemes for flood routing computations because the size of the time step used in the computation is restricted with the latter. Verwey's scheme is a relatively uncomplicated version of the tried and tested Preisamann'a scheme.

The model was developed on a Hewlett Packard HP9845T desktop computer but has been converted to run on the Hewlett Packard Series 200 and Series 300 machines as well. Running time on the HP9845T machine ia about 0,1 seconds per time step per cross- section; on the HP9816 machine it is about 0,18 seconds per time step per cross-section.

Data requirements for the model « e the river cross-sections, the inflow hydrograph and the initial conditions in the river, the latter being in the form of the flow and water level at each cross-section.

The model is also suitable for inclusion in e- ’"id or basinmodel as the channel routing component. A - models arebeing increasingly used by local authorities i v the purpose of determining floodlines for entire drainage systems sod, in addi­tion, provide them with the facility to ijtudy the effects of land use changes, canalisation and the introduction of dams on the system. There are obvious economies to be obtained where flood­lines are determined for the entire watershed by the local autho­rity concerned by means of a watershed model rather than on a piecemeal basis in individual townships.

Watershed models generally incorporate both the hydrological and hydraulic aspects of flood determination. The watershed is subdivided into subcatchments linked by channels. The hydrolo­gical characteristics of each sub-catchment are entered into the model such that for any given storm event an outlet hydro­graph from the sub-catchment is obtained. Starting at the upstream end of the uppermost reach the inflow hydrograph to the reach is routed through the reach using the streamflow

routing procedure. The outflow from the reach is then added to the outflow from the local sub-catchment to provide the inflow hydrograph to the next reach. The process is continued, adding any tributary inflows that may occur, until the total of all the routed hydrographs has been obtained at the watershed out­let. Reservoirs can be included in the model as reaches for which, if necessary, different routing procedures can be specified. In watersheds where there are a number of reservoirs in the drainage system, watershed modelling provides the only reliable solution technique for determining the response of the system to various storm inputs.

Heggen (1983) describes a watershed modelling technique which uses the Puls reservoir routing and the Muskingum channel routing technique. Other more eopfc'sticated watershed models are mentioned in Section 1.4.

In Section 1.2 of this report the various approximate flood routing procedures and their applicability are discussed. The so-called complete dynamic models are presented in Section 1.3 and in Section 1.4 the reasons for the selection of Verwey’s imp!1cit finite difference scheme are discussed.

In Chapter 2 the finite difference scheme of Verwey is described. The de St. Venant equations governing gradually varied flow for use with natural channels are derived, the discretisation of the equations is presented and the double sweep algorithm used in the solution and the treatment of the boundary conditions described.

In Chapter 3 the model is described with reference to the preparation of data and the selection of the time step and weighting coefficients. The effects of varying the time step weighting coefiicients are presented and the various forme of output are described.

The application of the model to two systems is described in Chapter 4. In the application to the Swartvlei estuary and lake system a tidal condition is used as the downstream boun-

dary condition and the results o£ the model runs are compared with measured date■ The second application is to a reach ofthe Rietspruit watercourse in the Transvaal, under conditions of the 50 year flood. The maximum levels reached by the flood are compared with those obtained from a steady flow assumption.

A summary of the report is given in Chapter 5, together with the conclusion.

1.2 A Review of Approximate Flood Routing Methods

The motion of a flood wave in a river channel is a form of gradually varied unsteady flow; the two partial differencial equations describing this flow as fist published by Barre de St. Venant in 1871 are as follows :

W.l)

Where A = cross-sectional area, u » mean velocity, h = depth of flow, g = gravitational acceleration, Sf * friction slope, S0 = bed slope, x = distance along the channel end t = time. In this form the equations do not include for lateral inflow to the channel.

Equation (1.1) is a statement of the conservation of mass in the system and is termed the continuity equation. Equation (1.2) is derived from considerations of the conservation of momentum and is termed the momentum equation or, more frequen­tly , the dynamic equation, since it is seldom a true statement of momentum conservation. These equations are derived in the form used in the model developed in this project in Chapter 2.

The de St.Venant equations cannot be analytically integrated because of their non-linear nature and numerical integration techniques offer the only solution procedure fur practical situations. While all flood routing models use the continuity

equation in a form similar to that given by equation (1.1) different groups of models can be distinguished by the number of terms considered in the momentum equation. Following the nomenclature of Weinmann and Laurenson (1979), models which retain all terms in the momentum equation are called complete dynamic models. These are discussed in Section 1.3. While being the most complete, thane models are also the most demanding on computet resources and approximate models which produce results at considerably less expense have been developed. On the other hand the approximate models are limited in their generality and accuracy and with large memory, high-speed desk top computers common in engineering offices the complete dynamic models are likely to become more popular. The group of models in which the inertia terms */g u/6t and u/g "S"/are omitted from the dynamic equations are called approxi­mate dynamic models, diffusion analogy models or kinematic models corrected for dynamic effects, depending on the form in which the equations are expressed. These models are discussed in Section 1.2.2, Further simplification Of the dynamic equa­tion results in the equation

Sf = S0 (1.3)

which forms the basis of the kinematic wave models, also used extensively in overland flow modelling.

By using a general channel flow formula such as the Chezy or Manning formula oE the form

Q = K/Sp (1.4)

Where Q » uA is the discharge and K is the conveyance, and by expressing the conveyance in terms of the uniform flow where Q0

Q o - K A T (1.5)

an expression of the following form is obtained.

9 " <b r

By substituting for Sf fro-a the dynamic equation, Meinmann and Laurenson (1979) obtain an equation for a looped rating curve which is useful for illustrating the differs ces between the various flood routing models :

i X - - S g 6x S g 6x

diffusion analogy complete dynamic w

A typical looped rating curve is shown in Figure 1.1.

M ax. w a t e r le v e l

• s t e a d y flo w r a t i n g c u rv e

lo o p e d r a t i n g

D is c h a r g e Q

FIGURE 1.1 Typical looped rating curve

1.2.1 Kinematic models

The combination of the continuity equation and Equation (1.3) yields the kinematic wave equation

in which the coefficient c, called the kinematic wave speed, can be determined at a particular croas-section at distance x

l che KleiCa-Seddon Law

Wliere Bs = dV d y ifl Che aurtace width at section x. The deriva­tive ^2) in this equation must be determined from the steady flow rating function in order for equation (1.7) to be equiva­lent to the combination of the continuity and uniform flow equa-

Cunge (1969) he* developed an explicit finite difference scheme for Equation (1.7) which introduces a weighting coefficient in the finite difference approximation of the tints derivative. The scheme is similar to the box schemes used in the implicit complete dynamic models, discussed further in Section 1.3.4. The kinematic wave equation is, applied at a point within the four point grid aa shown in Figure 1.2.

T im e t

( n + l ) A to r 1 Io m

AtQi

—or., A

i-e)Ax 6AxAx

A t / 2

FIGURE 1.2 Computational grid for kinematic models

Using thn superscripts ti and n+1 to denote the value of the variables at time level nit and (n+1) At respectively and the subscripts i and i+1 to denote the value of the variable at

distance iAx and (i+1 )Ax respectively, the discretised form of the equation is :

_L -<$ * ( ' - - C i ) + C \ - q-'1 + € \ -fi<c> fit 2 fix

= 0 d-9)

In which <■!> s ^ where <c> is the average value of c for the reach and is independent of time. By introducing the travel time parameterk * Ax/co this equation can be reduced to she classical Muskingum equation by McCarthy :

Qit! B Gl Qi + c < - + C5 qJ+1

V1 ’

and + C2 + C3 = 1

The value of k can be determined from observed floods or by means of Equation (1.8). The value of B can only be determined from observed floods, Since these values have not been deter­mined from physical considerations, extrapolation is risky.

The kinematic wave equation, equation (1.7), does not enable attenuation of the flood wave to be represented; the attenua­tion obtained with the Muskingum method is numerical, due to its being a poor approximation of Equation (1.7). Cunge shows that the Muskingum equation is also an approximation of the diffusion equation

Where D is the diffusion coefficient, if the parameter 6 is evalu­ated from

The diffusion equation allows for flood wave attenuation and relates it to the physical characteristics of the river. It is discussed further in Section 1.2.2.

By retaining the time derivatives in the kinematic wave equa­tion (1.7) and using finite difference approximations foe Che space derivatives, Koussis (1976) shows that equation (1.7) may be written as an ordinary differential equation with a Muskingum type solution given by Equation (1.10) but with coef­ficients

Koussis shows this to be a generalized kinematic routing model reducible to other well-known kinematic models, the Muskingum and Kalinin-Miljukov models, by appropriate selection o£ the routing parameters. In the flood routing procedure proposed by Koussis, dynamic effects are considered by introducing a looped rating curve of Che form given by the Jones formula,

BsSocAx (1.12)

Cl “ <oat (i - y ) -y ; C2 * * <c>at (i - y)

Where Y “ exp (- and Ci + Ca + C3 = 1 (1.13)

(1.14)

to give improved estimates of the travel speed parameter <c> for the sub-reach. The procedure is an iterative one since the travel speed parameter <c> is first estimated for the subreach and the

hydrograph is muted through the subreach. The stage hydrograph is then obtained for the downstream end of the subreach and used to compute the relevant segments of the looped racing curve at the downstream end of the subreach. The slopes of the loop- i rating curves then yield the improved ,=1ue of <c> for the reach.

Weinmann and Laurenson (1979) describe this model as a kine­matic model corrected for dynamic efffects and show that it gives the closest approximation to the complete dynamic model of all the kinematic models. Weinmann has developed a general computer program which allows the user to select the kiiwaatic model version and tht input and output options that best suit his application.

Peterson and Verhoff (1982) have developed an equation of the Muskingum type which is still more general than that used in the Koussis model (Equation (1.13)). The generality results because the coefficients were not obtained from an approxi­mation of a differential equation (Equation 1.7) but from mathematical approximations of the conservation of mass.

I’uang (1978) presents an alternative approach to the solution of the kinematic wave equation to those which reduce to the Muskingum form. By expressing equation (1.3) in the form

Q ■ (1.13)

Where et and m are routing coefficients whose values depend on the channel properties, the continuity equation (1.7) can be

The reach is divided into a number of sub-reaches for which the coefficients and m are assumed to be constant. These coeffi­cients are determined by application of the Manning formula to Che typical cross-section for the sub-reach. A method is given for determining the values of a and m from topographic maps

discharge and flow area is not available. Forlels, where the values of a and m vary with thesn appropriate set of routing coefficients areeach time level based on the flow area at the

1.16) ton-linear routing ire the following

(1.19)

By substituting these equations into equation (1.16), an explicit equation relating the unknown to the known valuesA i 1 • A ? and ia obtained. The solution thus proceedswithout iteration.

In the non-linear procedure, first developed by Li, Simons and Stevens (1975), the continuity equation is first discretized using equation (1.17) and

( 1 .2 0 )

before the following substitutions, obtained from equation

dr' - « ur'i"

resulting equation is non-linear in terms of the unknown ] and must be solved iteratively. Using the linear solution

L

as Che first approximation, Huang presents an iterative proce­dure which converges rapidly, three iterations usually being sufficient.

In a comparison of the non-linear and linear routing procedures Huang found that for time steps of 15 minutes the solution ob­tained using the two procedures agreed quiet closely. When a longer time step of one hour was used a narked difference between the two solutions appeared, that from the non-linear procedure being closer to the solutions obtained using the 15 minute time step. Huang thus recommends the use of the linear routing procedure with short time steps. The method is simple and has been succesfully incorporated into a watershed model.

1.2.2 Approximate Dynamic and Diffusion Analogy Models

Where the continuity equation and the indicated part of Equa­tion (1.6) are solved directly by the same numerical methods as used with complete models, the, resulting models are called approximate dynamic models. However, since these models require almost the same computational effort as the complete models they do not hold any advantage over the complete models except that they have simpler equations for the coefficients in the solution procedure and are thus easier to programme.

Diffusion analogy models, on the other hand, are those wherethe continuity and simplified dynamic equations are combined into a single equation of the diffusion type, expressed either in terms of flow depth or discharge. For example, written in terms of the discharge the equation is

Where c is the convection speed and D the diffusion coeffi­cient, both being constants. The diffusion method was derivedby Hayami (1951) for application to flooding in irregular river channels. The convection speed c in this equation is notstrictly a typical value of the wave speed averaged along the

reach and over a certain range of discharge; it depends also on the degree of attenuation in the reach. Tbi success of Che method depends on a knowledge of c a-.td r. for the reach; a disadvantage ia that these parameters can vary significantly with the magnitude of the flood.

In the variable parameter diffusion method developed by Price (1973) the parameters c a ad D are defined as functions of the discharge by correlating values of c and D calculated for a number of recorded floods vi th the average peak discharge along the reach in each case.

1.2.3 Applicability of the Approximate Flood Routing Methods

The relative importance of the various terms in the dynamic equation depends on the characteristics of the inflow flood hydrograph and the river system under consideration. In flood routing computations the irtertia terms are generally small and can be ignored without significantly affecting the solution.

For example, a typical reach of the Rietspruit has a bed slope of about 0,4%. A flood with a 50 year recurrence interval has a peak ot about 200 cumecs and results in values of the terms ~ 'If an,i "g fx o£ about 0,012 for a point midway on the risin, limb of the hydrograph, not significant in comparison with the bed slope. Figures given by Miller and Cunge (1975) for situ­ations where the inertia terms were expected to be significant show that in fact tbgy could have been omitted. The approxi­mate dynamic or diffusion analogy models would therefore seem to be sufficiently accurate for most flood routing purposes.

Hovever, in situations where the maximum water levels reached by the flood are of interest, omission of the convective accele­ration term ~ results in the incorrect representation of the backwater profile under steady flow conditions. The dynamic equation with the local acceleration term omitted can be

which, under steady state conditions, is the energy equation for backwater profiles. Thus by omitting the -j- term theinfluence of the velocity head on the steady state water sur­face profile is ignored.

In a comparison of the Muskingum-Cunge, linear diffusion and variable parameter diffusion methods under conditions typical for flooding in British rivers. Price (1973) concluded that the Muskingum-Cunge method is generally aa accurate as the linear diffusion method Ln all circumstances and because it is simpler conceptually preferred its use to .he linear diffusion method. He found that the variable parameter diffusion method has some advantage in accuracy where there is inundation of a large flood plain and when a sequence of floods with a wide range of peak discharges is being routed. However, the advantages of the variable parameter diffusion t-ethod ware outweighed in many cases by the more complicated computer programming required.

The discinguishing feature of the kinematic models such as the Muskingum-Cunge model is that the discharge is always equal to the normal discharge and is thus a single valued function of the depth. Backwater effects are thus completely ignored in these models. Weinmann and Laurenson (1979) show that for channels with well-developed loops in che rifting curve, errors o£ up to 25% in the computed peak flow can arise with the use of a kinematic model. They conclude that :

1. The omission of the acceleration terms in approximatemodels will not significantly effect the accuracy of theflood routing results in normal circumstances.

2. For slowly rising hydrographs and moderately steep channels a model of the kinematic type can be expected to give results that differ only slightly from che ones obtained form more complete models.

3. In channels of flat grade or vieh steep hydrographs, orboth, the pressure term ^ is significant and a kinematictype model should not be used.

By considering small sinusoidal perturbations to the equili­brium flow using linearised equations, Ponce, Li and Simons (1978) found the limit of applicability of the kinematic model, to be given by

Where T = wave period of sinusoidal perturbation, equivalent in practical situations to the flood wave duration, h0 = uniform flow depth, Uy = uniform flow velocity, t » dimensionleas waveperiod of unsteady component of motion. They found that for atleast 95Z accuracy of the kinematic wave equation the dimension- less period t has to be greater than 171. Thus for mild ch. -nel slopes the period has to be very long for the kinematic model to apply. Similarly the limit of applicability of thediffusion model was found to be

They conclude that the diffusion model applies for a wider range of slopes and periods than the kinematic model with the added advantage that the diffusion model does allow for physical attenuation.

For the example of the Rietspruit mentioned above, and which is discussed further in Chapter 4, the flood wave duration would have to be more than eight hours for the kinematic model to apply and more than one hour for the diffusion model to apply.

1.3 Complete Dynamic Models

The two methods in general use for the numerical solution ofthe complete de St.Venant equations are the method of charac­teristics, baaed on the characteristic form of the equations, and the finite difference methodo, which are based on the partial differential equations as derived.

(I.24)

(1.25)

1,3.1 The method of characteristics

Abbott (1975) describes the method of characteristics as a tech­nique whereby ths problem of solving two simultaneous pa.tial differential equations can be replaced by the problem of solving four ordinary differential equations.

The characteristic form of the de St. Venant equations are :

(l.M)

+ + c) (" + 2c) = -g(Sf - So)(1.27)

^| = u - c

(1.28)

^lt + H 1 - 2c) - -g (Sf - so)

(1.29)

T im e f

D is ta n c e x

FIGURE 1.3 Basis of the characteristic method

Characceristice can be defined as lines in the (x,t) plane along which disturbances propagate. The differentiation operators in equations (1.27) and (1.29) are the total derivatives along the characteristics. Equations (1.26) and(1.27) define the so-called forward characteristics and equations 1.28) and (1.29) defining the backward characteris­tics. The solution of the characteristic equations is obtained at the intersection of the forward and backward characteristics by integrating along the characteristics. Thus for any point Min the (x,t) plane shown in Figure 1.3 at which the forward andbackward characteristics from points L and R respectively inter­sect, the coordinates of point M and the values of the unknownflow variables uy and yM can be found from

* /t" dt"W " V "

"M " 2cK " "r " Z=K * /t“ =<So - Si,dt a '3°>

In taking integrals along the characteristics, no approxima­tion is made compared to equations (1.26) to (1.29). The approximation arises in the evaluation of the integrals, usually by means of the trapezoidal rule. Four non-linear algebraic equations are obtained which are solved by iteration. Since the value of the co-ordinates (x^i % ) obtained in this way must be determined for each intersection point a variable grid of points is obtained in the (x, t) plane aa is shown in Figure 1.4, This method is referred to as the variable grid

D is t a n c e x

FIGURE 1.4 Characteristics on a variable grid

Liggett and Cunge (1975) provide an algorithm for the solution of the characteristic equations with rapid convergence and a good degree of accuracy.

The variable grid method requires two kinds of interpolation when applied to natural channels. Firstly the geometric and hydraulic characteristics of the channel are known or defined only at a limited number of sections but are needed at any point of the (x,t)-plane. Secondly, the computed results must be interpolated to obtain hydrographs at any point or free surface profiles at any time.

Abbott (1979) describes the three-point method of characteris­tics, with which values of the dependent variables can be obtained at fixed grid intervals, and the four-point method of characteristics, the latter being generally superior to the three-point method for machine computation. However, Abbott and Verwey (1970) recommend the use of implicit finite diffe­rence techniques rather than the method of characteristics for flow in natural watercourses.

According to Cunge, Holly and Vetwey (1980) the originalvariable grid method of characteristics is not widely used for industrial modelling because of its complexity and the need to interpolate. Characteristics on fixed grid are employedsomewhat more frequently but the method is complex and more costly than finite difference methods, without offering anyimprovement in accuracy.

However, the characteristic method does have two important fields of application. Firstly, it is used as a standard for other methods, since its solution may be brought as close to the true solution of the basic equations as is required.Secondly, it is used as a means of representing boundary conditions in methods which cannot compute all flow variables at exterior or interior boundary points of the model.

1.3.2 Finite Difference Methods

In finite difference methods the differential equations of flow are replaced by algebraic finite difference relationships which are solved at a finite number of grid points in the (x,t)-plane termed the computational grid.

There are two basic types of finite difference schemes. In theexplicit schemes the equations are arranged to solve for onepoint at a "ime, generally making use of the values of thedependent variables determined at the previous time step at a few adjacent points. In implicit finite difference schemes a system of equations at the new time step involving all thepoints in the reach is solved simultaneously.

Liggett and Cunge (1975) describe in detail a number of diffe­rent schemes of each type, namely the Diffusive, Leap-frog, Lax- Wendtoff and Dronkers explicit Schemes and the Preissman, Vasiliev and Abbott-loneacu implicit schemes.

1.3.3 Explicit Finite Difference Schemes

The explicit finite difference schemes are the simplest: Co program, bud are subject to limitations on the size of the time step used in the computation, as is discussed further below.

£n the Diffusive or Lax Scheme the discretization of the derivative is as follows

With 0 = 1 this scheme is unstable; with i, = 0 the scheme repre­sents the so-called diffusive scheme.

Using this discretization in the de St. Venant equations the solution at the point (i, n+t) for depth cornea immediately from the continuity equation and for velocity from the dynamic equation. It can be seen from equation 1.31 and Figure 1.5(a) that the solution at the point (i, n+i) is dependent on that at (i-1, n) and at (i+1, n). The solution at the boundary points must therefore be determined using the characteristics method.

In the Leap-frog scheme centered difference approximations are used in both distance and time i.e.

(1.31)

<1.32)

« ■

The Leap-frog method is probably the earliest one ever used for one-dimensional flow modelling. It produces values of the un­knowns at all points in the grid and is of second order accuracy. A disadvantage with the method is that the solution obtained is a saw-toothed line since the points where the depen­dent variables are computed are alternately odd and even, the solution at the odd points being independent of that at the even points. The discretization for two adjacent points is shown in Figure 1.5(b).

The Lax-Wendroff scheme was first applied to the open channel unsteady flow equations by Houghton and Karahota. It is a second order scheme which is not dissipative - it does not smooth out initial perturbations in the flow and the initial conditions provided must satisfy the flow equations as closely as possible. The scheme is described in detail by Liggett & Cunge (1975) for application to a trapezoidal channel.

T im e ti

( a ) ( b)

z L \r r

i - l i i * l i -1 i i * l i+ 2

D is ta n c e x

FIGURE 1.5 Explicit discretization schemes (a) Diffusive (b) Leap-frog

2 n + 2 — C o n t in u itye q u a t io n

Zn + ID y n a m ice q u a t io n

2 n -

21-2 21-1 21 2i+l 2i+2D is ta n c e

FIGURE 5 Explicit diacretization schemes (a) Diffusive <b) Leap-frog (c) Dronkers

In Dronkec's expliciL scheme, developed mainly Eor tidal flow computations, different discretizations are used in the dynamic and continuity equations.For che dynamic equation

and fiu 2n+1 - i

while for the c itinuiey ettuation

u2i+2 ~ U2i+1 5 U “ >21^2 '(1.34)

It can be seen from the above discretizations, shown graphi­cally in Figure 1.5(c) that the difference equations cover three time-steps, from level (2n-l)At to (Zn+2)it, and four distance steps, from (2i~2)Ax to (2i+2)Ax. It is obvious that this scheme is valid only when flow variations and river geometry variations are slow and gentle in space avid time.

The major disadvantage wich explicit: schemes is that Che time step used in Che computation ia Limited by the Courant-Friede- richs-Lewy condition for stability :

[uo + ~ < 1(1.33)

Huang and Song (1955) describe a second instability in the characteristic, diffusive and leap-frog schemes given by the Koren equation :

at <_ /l+2 1

where u0 and c0 are the initial velocity and speed of the shallow water wave respectively. They found that this stability criterion can be increasingly relaxed the more the energy lo«- term is treated implicit/. The Koren stability criterion becomes very restrictive when the iiroude number, given by • is small.

The combination of the two stability criteria given by equatiou 1.35 and 1,36 defines the zones of stability and instability in the selectK.n of at and a x shown in Figure 1.6,

A tT im e s t e p

U n s ta b l e

A t m a x ,c o n d i t io n

S t a b l e

A xD is ta n c e s t e p

FIGURE f. 6 Zones of stability and instability for explicit schem«

In explicit finite difference schemes special attention must be given to determining the v®iue of the unknown dependent vari­ables at the boundaries. The method of characteristics is the only general technique for finding these boundary values. The general rule for the number of conditions specified on the boun­dary or initial line is chad it must equal the number of charac­teristics originating on the boundary or initial line. Obvious­ly two conditions will always be required along the initialline. In supercritical flow situations two characteristics ori­ginate at the upstream boundary and two upstream boundary condi­tions must be specified; no downstream boundary condition is re­quired, the solution at a point on the boundary being deter­mined from the characteristics intersecting at the point.

In subcritical flow situations the forward characteristic origi­nates at the upstream boundary and the backward characteristic at the downstream boundary. One boundary condition must there­fore be given at each boundary and the other determined usingthe equation for the backward characteristic at the upstreamboundary and the forward characteristic at the downstream boundary.

1.3.4 Implicit finite difference methods

The implicit methods of finite differences were developed because of the limitations imposed on the time step using explicit schemes and have no limit on the time step, providing that a sufficient number of points on the hydrograph are obtained to define it adequately.

A number of implicit schemes have been developed; generally these differ in the discretization of the terms in the equa­tions. A system of linear algebraic equations is obtained for all computations! points which is solved at each time level,with the boundary conditions linearized in the dependent vari­ables closing this system. Generally the matrix of coefficients is =- ?. with non-zero elements banded on the diagonal and£.<’■ ids of solution such aa the double sweep methoddescrioed in Section 2.2 can be useu.

The Preissmann, Amein and Verwey schemes are so-called four- point: or box schemes with the flow equation applied at a point centred within the four point grid as shown in Figure 1.7.

Jit M T i-e[ ©

AXi

D is to n c '- xFIGURE 7 Four point grid for box schemes

At the point M the average value and the partial derivatives of a function £ are expressed by

f(M) » (1 - *)(1 - 6) f" + (1 - |)e f f 1 + *(1 - 0) £° + we?

A feature of the box scheme is that centred difference approxi­mations are used for both time and space derivatives whereas other schemes use combinations of forward, centred and backward difference approximations. In the scheme of Amein and Fang (1970) 8 ■= i|j = 4. Generally in Che Preisemann and the Verwey schemes i|i = 4 and 4 5,'8< 1

In Che Pteissmann scheme the discredaed £low equations areLinearized in terms of 6y and AQ by putting

yi+1 “ yi Ayi

( 1 .3 8 )

<{ " ( ( + AQ.

and developing the terms of the equations by means of power series expansions with second and higher order terms neglected. The continuity and dynamic equations at each point reduce to the form

W i + ■ ci "i * V qi * Eiand (1.39)

" m . , * " * 'iWith the boundary conditions linearized in y and Q the system of equations is solved at each time step. Using the known values of the previous time step the coefficients to can be computed and the set of equations solved, providing the second approximation to the unknowns. The significant feature of the scheme is that in most cases the second approximation is sufficiently accurate and thus only one iteration is required at each time step.

In Verwey's scheme the equations are written in terms of ,

Q ^ , and and are of the form

Ai Yf 5 * Bi < 1 * v K l + h€\ “ Bi

' i f * * ' i f * h f ' h

In the first iteration the coefficients are evaluated by approximating the values of and Q?*1 with the knownvalues y” and Q^ . The coefficients are then adjusted using the values of Y?+ ' and Q?+1 obtained from the first iteration

and the second iceracion carried out. As a rule two iterations per dime step are needed to give a sufficiently accurate simulation.

in the scheme of Arnein the equations, in terms of y and u, are

. ( f • c

The solution proceeds by assigning trial values to the un­knowns, usually the values from the previous time step. With these trial values the right hand sides of the equations willbe non-zero, acquiring values known as the residuals. Theresiduals and the partial derivatives of Equation (1.41) are related according to the generalized Newton iteration method by Sf. 6F. <SF. 6F.

dy. + dO. ♦ dY.+1 ♦ dU. + 1 - Ru .

i 60. SY.+1 6U£+j i+1 "2,4Where Rj, i and R2, £ are the residuals associated with the functions Ft and Gi respectively and

dYi = Yi- K+f - = dUi ” Ui, £+t - - V(1.43)

Where ^ and are the values of the unknowns after thekth iteration.

The system of equations (1.42) are solved and revised values for the unknowns are obtained and the procedure is repeated until the difference in values of any unknown in two consecu­tive iteration cycles falls below a tolerance limit. The number of iterations required to provide a reasonable solution is not stated.

Other implicit schemes which are not based on the box or four- point method are the Vasiliev scheme, the Abbott-Ionescu scheme, the Delft Hydraulics Laboratory Scheme and the Gunaraensm- Perkina Scheme.

The Vasiliev and Gunaratnam-Perkins Schemes are similar to the box schemes in that the values of the dependent variable are determined at each grid point. The discretization is different, however.

In the Vasiliev Scheme the discretization of the flow equations

The scheme is applied to the cone form but with the dynamic equation

mity equation io its usual 1 the following form

Where c = /gg and * = resistance and slope terms. For N compu­tational points and 2N unknown dependent variables a system of 2N - 4 equation is obtained. Two boundary conditions and two characteristic equations written for the limit points i = 1 and i = H close the system, which is then solved using the double sweep algorithm.

The GunaraCnam-Perkins achtsme links together three consecuti the discretization

a fully implicit scheme which points i-1, i and i+1 using

. 4( £ f ' - « ? ) / a t - f= ) / i t

As with the Vasiliev scheme, two boundary conditions and two characteristic equations are required to close the system of equations.

The Delft Hydraulics Laboratory and AbbotC-Zonescu schemes are similar in that the computational grid is divided alternately into "y-points" at which water stages are computed and "Q-points" at which the flow is computed. The Delft scheme is based on the concept of computational cells at the centre of which the water stages are computed and which are. linked to adjacent cells on the left and right through discharge laws. In both schemes the space derivatives ate the weighted mean of those at time level nit and (n+1) it using the weighting coefficient 8 ;

II ■ * O / * - * 0 "61C£it| "

where O,$<0<1,O in the Delft scheme and 6= 0,5 in the Abbott- lonescu scheme.

lu the Abbott-Ioncscu scheme the coefficients of the discre­tized equations are evaluated at time level (n+i)At, requiring an iterative solution procedure. In the first iteration the valu' of the coefficients are evaluated from the flow varia­bles at time level nAt. The set of equations is then solved using the double sweep algorithm. In the second iteration the coefficients are evaluated at time level (n+i)&t using the values of the flow variables at (ndt) and Chose at (n+IMt obtained from the first iteration. As for Verwey's scheme, two iterations per time step give satisfactory results,

1.3.5 Comparison of explicit and implicit schemes

Price (1974) compares the accuracy and efficiency of the Leap­frog explicit method, the two-step Lax-Wendroff explicit method the four point implicit acheme of Amein and the fixed mesh characteristic method, using a monoclinal wave for which the exact analytical solution of the full equations is known. The monoclinical wave bears a strong resemblance to a flood wave and in very long rivers the front of a flood wave may take the form of a monoclinical wave. Price found the explicit mmthods became unstable when the time step exceeded that given by the

Cournnt-Friederichs-Lewy condition and the characteristic method became unstable tor time steps in excess of ten times this value. No instability was recorded using the implicitmethod. In the comparison Price adopted a 40 metre wide pris­matic channel of bed slopes 0,001 and 0,00025, typical of British rivers, and of 100 km length. He used distance stepsin the range 2 500 metres to 20 000 metres and time-stepa inthe range 180 to 14 400 seconds. In all methods used thesmallest error was achieved for the smallest value of thedistance step,

He found that the implicit method is markedly more efficient than any other method for a similar accuracy, the reason being that the least error was obtained with the implicit method for greater time steps.

Chaudhry (1979) lists the advantages and disadvantages of explicit and implicit finite difference schemes as follows :

1. Stability - Explicit schemes are conditionally stable; the Courant-Friederichs-Lewy condition must be satisfied. Implicit schemes are unconditionally stable.

2. Ease of programming - the explicit method is easier to programme.

3. Economy - with the larger time step allowed with implicit methods less computer time ia required in the computations. However, seen in the brooder context of the total cost of preparing the model and analysing the results, this aspect is not always important.

4. Computer memory requirements - usually more storage is required in an implicit method than in an explicit method.

5. Simulation of special cases - where conduits have closed tops, eg. in stovmwnter or sewer system and tailtace tun­nels, the free surface width can become very small and the size of time step must be reduced accordingly - the impli­cit schemes should be used in these cases. On the other hand implicit schemes usually fail to represent supercriti­cal flow so that when the Froude number becomes greater than one during computation the results may be unstable.

6. Simulation o£ sharp peaks - because of Che smaller Cime step explicit; medhods are generally more suitable. With the same size of time step computational time with implicit schemes would be greater.

7. Formation of bores and shocks - the explicit schemes are more suitable for the analysis of transients in which a bore forms.

2.4. Selection of lilood routing method

the selection o£ the flood routing method used in the model has been based on the requirement that Che model provide both the Stage and the H o w At all points along the reach. A secondary requirement is that the model be suitable for inclusion in a watershed modelling system as the channel routing component.

the MOPSET watershed model described by Huang (1978) uses the kinematic method for routing floods down natural channels. The kinematic method is the simplest of the flood routing methods but its range of applicability is limited. Since it is based on a single-valued stage-discharge relationship at all points in the reach the kinematic method cannot reproduce the dynamic effect of flood wave propagation nor any backwater effects in the reach.

Akan and Yen (1981) provide a method for routing floods through channel networks using the diffusion analogy model. In a comparison with the complete dynamic and kinematic methods applied to the same channel network, they found the diffusion analogy model to be nearly as accurate as che complete dynamic method and faster and cheaper in computation than the kinematic method. However, although the diffusion analogy model will simulate the backwater effects at channel junctions etc., its exclusion of the convective acceleration t e r m - g c a n result in inaccuracies in the water surface profile obtained, especially in natural channels where differences in the velo­city head at adjacent sections can be significant.

McMahon, Finzgerald and McCarthy (1984) describe the BRASS (Basin runofE and streamtlow simulation) model which incorpo­rates the complete dynamic channel routing mooel developed by Freed (1978). The model is capable of simulating lateral in­flows to streams and dendritic (branched) river systems. The disadvantage with the use of complete dynamic routing models in a watershed model is that the dynamic routing models are large programs requiring considerable core memory allocation for execution. Watershed models such as BRASS are therefore divided into segments with a root segment which resides in memory throughout a simulation. A second disadvantage is that the complete dynamic models are generally somewhat slower computationally chan the approximate models. The models based on explicit finite difference schemes are slower because of the limitations on the time step imposed by the Courant-Lewy- Friedrichs condition, while those based on implicit finite difference schemes are slower because of their greater comple­xity. However, with the rapid advances being made in computer technology and high speed 32-bte computers becoming increasing­ly available in desk-top form, these disadvantages are not considered significant.

The model developed in this project has therefore been based onChe complete dynamic form of the de St. Vanant equations. Animplicit finite difference scheme can be seen to have advan­tages over an explicit scheme for application to a general pur­pose flood routing model where the flow is generally subcriti- cal in that relatively large time steps can be used in the computation. The Preissmann scheme has been extensively used in industrial models. The formulation of the coefficients used in the scheme is extremely complex, however, and Verwey'svariant of the scheme has been used for this project.

Although the model developed in this project is for application to single channel reaches, the solution technique can beapplied to dendritic channel networks after some modification.

Cunge, Holley and Verwey (1980) and Joliife (1984) describe the application of implicit finite difference schemes to channel networks.

/ : , b

2. DESCRIPTION OF THE FINITE DIFFERENCE SCHEME OF VERWKY

The de St. Venant equations are derived for natural channels of irregular cross-section in Section 2.1 of this chapter. The discretization of flow equation* ia presented in Section2.2 and the double i*i. ep algorithm used in the solution of the set of equations obtained from the discretization is presented Lrt Section 2.3. Finally, the treatment of the boundary conditions is discussed '■ Section 2.4.

The da St. Venant Equatic Natural Channels

> For One Dimensional Flow i

The de St Venant equations are based the following hypo-

the velocity is the water level

ind vertical pressure is

i) The flow is one-dimenclonaluniform over the cross section and across the section is horizontal.

ii) The streamline curvature is smai'accelerations are negligible, hence hydrostatic.

iii) The effects of boundary friction and turbulence can beaccounted for through resistance laws analogous to those used for steady state flows.

iv) The average channel bed slopei is small so that thecosine of the angle it makes with the horizontal may be replaced by unity.

The co-ordinate system and definition of terms used in this report are shown in Figure 2.1. Two physical laws are used to derive the equations for unsteady flow in open channels,conservation of mass and conservation of momentum.

With reape-t to the control volume shown in Figure 2.1 the law of conservation of mass can be formulated as follows :

Figure 2 Definition Sketch

NeCt mass inflow into control volume = change in mass storage

pQAC - p(Q + r^Ax)At - p(A + ~At)Ax - pAAx

Expressing this equation in terms of water level by using

M . i i i i6t 5y 6tand putting Bs = ^ the continuity equation is obtained.

The momentum equation can be derived by equating t-bs-sum of the nett rate of momentum entering the control volume and- the forces acting on it with the rate of change of momentum within the control volume.

Net rate of momentum entering the control volume

• Q» - (Qu + «x)

Three types of forces are considered to act on the control volume, namely gravity, pressure and friction.

Fg - pgAAx sinx = - pgASo

?P = \ ~ CFp + 5 ^ x> ” " 5^BAs

= ^hD g (h (x )-n )c i(x ,n )d n A x

/ h 5= -pgA x (h (x )-n )< 7 (x ,r t)i iii

= -p g A x |~ ^ ^ o ( x ,n ) d n + ^ (h - ( x ,n ) <

= - p g A x | - | | + J- (h - n ) ^ Cx, n )d n |

The second cerm on the right hand side of this equation repre­sents an increase in the pressure force as a result of a change in the width of the channel; in prismatic channels it is assumed to be zero. In non-prismatic channels as the channel widens or narrows the banks contribute an additional pressure force which exactly cancels this term.

Thus Fp * -PgA

Friction : F£ = pgA SfAx

where Sf is the friction slope.

The rate of change of momentum within the control volume can be written as

Rate of change of momentum (pQ)Ax

Combining these elements gives the momentum equation

-p~r~Ax - pgASoAx - pgA^Ax + pgAS-Ax

= "gf (DQ)Ax

Dividing through by P and Ax and rearranging gives

When working withthe dependent

Futhermore the velocity u can be expressed as Q/A.

In addition a momentum correction factor E , called Boussinesq coefficient, should be applied where working compound cross-sections with non-uniform velocity distribute

(2.2)

>£ depth-averagedWhere the subscripts Z denote local velocity and depth at position Z in th

With these changes the momentum equation becomes

In this form the equatv 1 dynamic' equation since

generally

The continuity and momentum equati- expanded to include for lateral in form of a distributed Inflow to the channel frc groundwater sources, concentrated inflow at I

presented

Insofar as the continuity equation is concerned, for a distributed inflow of q per unit of channel length a volume of qAxAt enters the control volume during the infinitesmal period At,

The continuity equation thus becomes

To the momentum equation must be added a term to account for the additional momentum both entering and leaving the control volume. The additional momentum entering the control volume is pqUgAx where Ug is the downstream component Of velocity of the lateral inflow. Upon leaving the control volume the velocity of the additional flow is the same as the channel velocity, so the momentum leaving is pqu £x. Some authors (Liggett [1975]) and Cunge, Holly and Verwey (1980) include this latter term as a separate term in the dynamic equation. However, examination of the continuity equation including lateral inflow shows that the flow leaving the control volume includes the inflow to the control volume. The term representing the nett rate of momentum entering the control volume, —' Ax , will thusinclude for the momentum of the lateral inflow leaving the control volume and the momentum equaticr becomes ;

% » n + * % - 1 " , - - o

Other forces can be included in the momentum equation besides gravity, pressure and friction forces. Wind effects can be included where these are significant, for example, in the model of th. Swartkops River and estuary in Vort Elizabeth developed by the National Research Institute for Oceanology (Huizinga (1984))

The model developed in this project has been based on the Equations (2.1) and (2,3) derived above, without consideration

-43-

2.2 Discretieacion of the Flow Equations

Writing f(x,t) = ffulx, n<St) « for the general form of the dependent variables at the point (iAx, nAt) in the computa­tional grid che discretieation of these variables and their time and space derivatives follow the Preissmann scheme :

- (i - e>(l - + e d - lo t”* ' t (i - e ) t * 6*£"J

# - e<f"! - f * (l - e)(f” , - A / m

If -'('% - "I.,)/" * " - +)(?' - ».*)

where 9 and f are the .:ime and space weighting coefficients " respectively, as depicted in Figure 1.7.

With the St.Venant equations as derived in Section 2.1

# + 0.1)

o.3)

the mass or continuity equation is simply discretized as follows ;

' ( < 1 1 - + c - ' ) ( < „ - * ' ' M

^ Cyi+i ■ yi+v /tic * Bsr 9 o ~ ^ r 1 ■ “ °

BBs"*' + 11- 6)Bs”

Grouping the dependent variables at time step fl+1, i.e. the un­knowns, on the left hand side of the equation, multiplying through by Ax^ and rearranging, an equation of the following form is obtained,

I ' l o T * " m " ' + " I ' w ' " i(2.6)

Ali = - 6

Bi^ = Bsi+e C* ~ 1)Ax^/At

Cli = ® (2.7)

Dl^ ^ Bsitl

E H ” Cl - S X Q ^ - Q ^ ) + 8 1 ^ + 0 1 ^ ; + ,

Before discretizing the dynamic equation as given by Equation (2.3) it is necessary to convert the friction slope to a relation between friction losses and discharge of the type

Where K = K(x,y) is the conveyance of the channel. The value of K can be determined from any of the well-known empirical resistance laws such as Manning or Chezy, Expressed in terms of the Manning equation

K - — (2.8)

The dynamic equation can now be w 1tten as

U.l)

of the dynamic equation is complicated by

the presence o£ the non-linear e In Verwey's scheme, the discteti; to that used in the Abbott-Ionescu scheme,

(#4of these terms is similar

(2 .10)

Slal,

.n+i

Since effectively Q ** |Q?| corresponds to the square of the geometric mean of the discharge at the two time levels, the value of K used in the model has been taken to be the geometric mean of K® and K rather than the arithmetic mean used by Votwey. Thus, in the model the following discretization

The different discretii the value of K?-1"6 given by Verwey is depends. .

is only significant if 6 > t since

The discretized dynamic equation thus becomes

' " ^ 1 ' ' ' ' S ' / ' " '

- 1 * *5 k a - "

- » ; ] t o i * i ‘ K i [ K l l K * , i / c " ' k u i > * i ' - *> _Qjl/tK?*1 Kj)J - 0

(y?

<' the mass equation, the dynamic equati<

* 2 ^ » »2.y. » 0 2 ^ , + 02^,^, - E2. (2.14)

A2{ - (1 - 6"teg°/6f e * giXjd -♦) iJiJIgJI/tKf K?)

H i ' '

H I - 4 H .1

(2.15)

» ( - 4 4 , / i t * ( J - H Q ?) * (1 - 6 )e « ? ‘ 5 <»“ * l - y” >

The coefficients Al ^ Bli, etc, and A2{, B2^, etc. are known functions of flow variables. Equations (2.6) and (2.14) are two linear algebraic equations in terms of q !?+' , q"*| andY?*|for every pair of points (i, t + 1). For N computational points there will be a system of 2N-2 equations for 2N unknowns.

With the addition of the boundary conditions the system can be solved using the double sweep algorithm.

It can be seen that the equations for the coefficients, Equation (2.7) and (2.15), include the values of the flow variables Ba, 0 , A and K at time level (n+i)4t. The flow equations must therefore be solved Iteratively at each time

step. In Che first iteration these flow variables are approxi­mated with their values from the previous time level, n t; in the second iteration the solution is improved by using the results of the first iteration. Two iterations per time step have been fouaj to give a sufficiently accurate simulation in most situations.

2.2 Double Sweep Algorithm

According to Strelkoff (1970) the double sweep algorithm has been in use as a means of solving the <1e St.Venant equations since the early i9601s. The method takes advantage of the fact that the only non-zero values in the matrix of coefficients lie in a band along the diagonal, as can be seen below.

With the boundary conditions linearized m y and Q the system of equations for a single channel reach has the form

(2.16)

(2.17)where I;

(2.18)

the left and right hand side boundary conditions reapec- The double sweep algo­

rithm uses the banded matrix structure of the linear system Of equations to compute Else solution, with the number of opera­tions proportional to N.

Assume Chat there is a linear relationship of the type

Q?*1 - ?iyj+l + Ci (2.19)

which holds at point (i,n+i) in the computational grid

Lf this is true it can be shown that an analogous linear rela-int, ( 1 + i , n + J ) bo Chat:

Q R f - * u i y f j ,1 * ° U 1 <2- 20)

This is done by subsituting equation (2.19) into equations (2.6) and (2.14).

<»U * * C11< H + "‘i C l 1 - E1i - “ i S ,2'2l)

(02^ + AZ.p^y^+i + C2tqJ+} + 02iyj+j - EZj, - A2iGi (2.22)

From equation 2.21 a relationship between y"+* and the depen­dent variables at point i + 1 i.s obtained ;

» r ‘ - - jqmpr “S t - m p A iT f. »Ji{ * ,i; ♦ atJfJ'Equation (2.23) can be written as

yP*i « H^QP+l * ltyj+} t Ji (2.24)

Cl. Dt. E1.-A1.G.k - - •sriATpT i li ■ i Ji - <2-24>

Eliminating yj+i> from equations (2.21) and (2.22) gives

Qj;{ <01 i (Hi * A2iFi) - * yfj}

<Dlt(B2[ * * Al^i))

- (Eli - Al10l)(»2l t A2iFl) - ( « r A2iGi)(Bll * AljF^) (2.26)

Dividing through by (Bl; + AljFj) and putting e ( = (B2{ + AZ^F^)and expressing Q1}*1 as a Cunccion of yH+l one obtains

m i - yn+l + ! W r V i (2.27)(e.H^CZ.) i+i cTiT ^ CIT

which is relationship o£ the form indicated by equation (2.20),

Thus it is shown that, IE the relationship Equation (2.19) exists af any point in the model, similar equations can bewritten for all the following points. By linearizing the boundary conditions in this form a set of equations for all points in the model can be obtained.

In the forward sweep of the double-sweep algorithm the values of the coefficient's »>•* determined at all points in the model as follows. The sweep is ineialisad by determining the coeffi­cients F| and Gj at the first point from the boundary condi­tions. By combining Equations (2.17) and (2.19) one obtains

= L^ i G 1 ” <2'29)

The values of , 1 and are then computed from Equation (2,25) and F% and C?2 from Equation (2.28). The algorithm proceeds in this manner from gvidpoints 2 to N as is shown in Figure 2.2(a). To initialise the return sweep at the right hand boundary, use is made of the.relation combining Equations (2.18) and (2.19), namely

_________ (2.30)R, + M u

The values of Q^+l and y{£j can then be determined from Equations (2.19) and (2.24) respectively and the return sweep continues with the values of Q-L end y| being determined at each successive gridpoint as is shown in Figure 2.2(b).

•Equa t io ns u se d ii fo r wa rd sweep

( 2 . 2 9 )

•E qu a t io ns u se d in re tu r n swee p

( 2 . 3 0 )

Y N -2

,2 Schematization of the double sweep algorithm (a) Forward Sweep (b) Return Sweep

only applicable to subcritical i be handled with a single-sweep conditions beinr given at the

The double-sweep a

ilgorithm, with

ipercritical through rly horizontal flow, the used in a single compu- s described by Abbottof comput

(1979),

2.4 Initial and Boundary ConditionIn order to solve the system o£ simultaneous equations in y and Q at time level (n + 1)61 it is necessary to know the values of y and Q ar all potties ac the previous time level (ntit), together with the inflow to reach, stage at the down­stream end of the reach ;?g+l ■

To start a flood routing Deputation it. is thus necessary to define the inital conditions, in the reach in terms of Q and y at all points in the reach.

The boundary conditions are generally specified in the form of an inflow hydrograph at the upstream end of the reach and a stage-discharge or stage-time relationship at the downstream

The values of L] > Lg and in the equation for the upstream boundary conditions, Equation (2.17), can then be determined from the ordinate of the inflow hydrograph ac the applicable time level as follows :

Lt = 1 | 1-2 = 0 ; L3 « Qf*1 (2.31)

The values of and G| required to initiate the forward sweep can be obtained from equations (2.29).

Where for the downstream boundary condition the stage is given as a function of time, the values of R p Rg and Rgin equations (2.18) can be obtained directly as follows :

»1 - 0 ; 82 - 1 : l3 - (2.32)

The values of and are known, having been determined in the forward sweep, and can be determined directly fromequation (2.19).

Where the stage is given as a function of the flow it isnecessary to obtain a linear approximation of the function. For example, a weir-type formulation

Q - Y(, - ,„>3/2 (2.33)

where yw is the level of the weir crest, can be linearised by using a first order Taylor series development :

Q f 1 - Qjj + («pn (yft+1 - yg) (2.34)

where ■* Y(yu - Y„) ^

Comparing this equation with Equation (2.18) gives

Rj = 1 ; R2 = - I Y(yg - yw)^ ; R3 = | y§

By specifying critical flow conditions at the downstream end of the reach the equation for the Froude Number is set equal to

-

and solved iteratively with the equation

o r 1 ■ w ♦ %

This has the disadvantage that the speed of the computation is slowed while these equations are solved; the alternative is to provide the information as discrete points in a rating table. The linear approximation to the rating curve in the region of interest will then provide the values of the coefficients.

3. DESCRIPTION OF COMPUTER PROGRAM

The computer program comprises a data entry section, the main computational section and av. output section and is described under these headings in Sections 3 I, 3.2 and 3.3. The user instructions and a listing of the program are given in Appendixes A1 and Bl respectively. In the form presented inthe Appendix Bl the program has a capacity of 100 cross-sections of 20 points each and 5 000 time steps. The programme capacity is purely dependent on the memory size of the hardware end can easily be altered to suit.

3.1 Data Input

The input data required for the model are the cross-section data describing the reach, the boundary conditions and the initial conditions in the reach. The boundary conditionsusually are of the form of an inflow hydrograph at the upstreamend of the reach and some description of the stage at thedownstream end.

3.1.1. Cross-section data

Cross-section data for natural channels is arranged in tabular form with the distance along the cross-section from some arbi­trary starting point, the ground level and the Manning rough­ness coefficient given for each point in the cross-section. The data file containing cross-section data is set up using a separate program. The user instructions for this program and a listing thereof are given in Appendices A2 and B2 respe­ctively. The data file structure is Che same as that used fora conventional backwater program, the principle difference being that the order of cross-sections required for the back­water program is the reverse of that required for the model.

' he advantage of having the structure of the data file the same as used in a backwater program is that steady state initial conditions for the reach can be determined and stored using the backwater program. This aspect is discussed further in Section3.1.3. A sample from a cross-section data file listing is given in Table 3.1.

REACH NUMBER 2

2 100,00 5 .8 0 ,0403 .4 5 .6401 .80 .0401 .80 .0402 .8 5 .0403 .4 0 .0405 .0 0 .040

3 160.00 5 .0 0 .0253 .5 0 .0252 .2 0 .0252 .2 0 .0255 .4 5 .025

0.01 1 . 0

TABLE 3.1 Sample listing of cross-section data file

It is not necessary for the cross-sections to be spaced at regular intervals along the reach; any spacing which adequately describes the physical configuration of the watercourse is acceptable. Generally for floodline determination the distance step should be of the order of 50 m to 200 m, depending on the flooded width.

Reservoirs can be included in a reach without special treatment provided that the spillway is given as the most downstream section if it operates under free discharge conditions; its discharge characteristic is then given as the downstream boundary condition. Gunge, Holly and Verwey (1960) describe methods oZ treating weirs as internal boundary conditions in a reach. This involves special treatment oE the coefficients at the section and the facility cannot be included in a model for general application. Where a reservoir or weir occurs in a reach the preferred method is to divide the reach into two with the reservoir spill, *V"or weir forming thn boundary between the two reaches. The outllow from the upstream reach then becomes the inflow to Che downstream reach.

The most important characteristic of a reSo'fvoir besides its outflow characteristic is its surface ares to water level relationship. Since the wat=v level slope in most smaller reservoirs is negligible it is not necessary to accurately represent the ground profile beneath the spillway crest. The cross-sections selected to define the basin should provide a reasonable representation of the surface area to water level relationship above the spillway crest.

The most convenient source of cross-section data is large scale topographic mapping (1:1000 to 1:2500) with contours at 0,5 or 1 ra intervals. This type of mapping is frequently available in municipal areas. Field surveys should be carried out to define hydraulic controls such as weirs and drainage structures in the watercourse as well as to determine the channel deptu •"■i. peren­nial rivers.

For larger rivers, where the objective of the flood routing exercise is not to determine the maximum flood levels with any accuracy but to obtain information on the passage of a flood, 1:10000 orthophoto mapping with 5 m contour intervals will often provide sufficient data for the exercise.

3.1.2 Inflow hydrograph data

The inflow hydrograph to the reach is usually defined by a number of discrete points on the hydrograph. With the size of the time seep given, the values of the inflow are interpolated at corresponding intervals. The hydrograph can be stored in a file for use in other runs. A number c-f inflow iiydrographs can be added in the model either directly or with a given time lt-j. This feature is useful where the outflow hydrograph from the upstream reach must be added to the run-oEf from the local catchment.

It is intended that the model be able to uao hydrographs gene­rated by hydrological models such as WITWAT or ILLUDAS when the facility for storing outflow hydrographs from these models is available.

Since the hydrograph is imposed upon a reaclji which is in an initial state defined by the initial conditions, it is necessary that the initial flow of the inflow hydrograph does not differ substantially from the flow in the reach in this initial state.

3.1.3 Downstream boundary condition

As was stated in Section 2.4 the downstream boundary condition can be specified by giving the stage as a function of the dis­charge or as a function of time.

A sCage-discharge relationship can be given in one of three ways. The first is merely to specify that critical conditions occur at the downstream end of the reach. The second is to give the coefficients of a weir-(:» i such as Equation(2.33). In the third method t jints on the ratingcurve are given.

The most common form of sta^e-cime relationship used in one- dimensional modelling is the tidal equation. The tidal equa­tion used in the model includes only the semi-diurnal lunar and solar components o£ the tide.

y = y + AM2sitt * AS2 sin ((2%(T + *)/TS2 (3.1)(TM2'

y = datum tor mean sea levelAm 2 = amplitude of the semi-diurnal lunar component (M2)Agg " amplitude of the semi-diurnal solar component (S2)tM2 " period of the M2 component (= 44 714 a)TS2 = period of the S2 component (» 43 200 s)$ = phase difference between M2 and S2 components (= 0 at new

A constant downstream water level can be handled by putting the amplitudes of the tidal components equal to aero.

3.1.4 Initial conditions

In order to initialise a run the model the initial condi­tions in the reach must be given ; the form of the water level and discharge at each point itt the reach. Obviously this initial state should be as consistent as possible with the flow equations in the model to minimise ‘•"tie amount of computer time spent in stabilising the system prior to imposing the flood hydrograph on it. The moat direct way oE achieving this is to furnish an initial condition as a backwater curve for a steady state condition with the flow equivalent to the base flow in the river or some fraction of che peak flow.

It is atill necessary to run the model under steady state condi­tions by holding the boundary condition constant for a number of time steps to allow initial perturbations to dissipate or

as given by equation 2.13 reduces to the discrete form of the energy equation by putting

* 8A<yi+i - r£)/4»t

Dividing through by and putting (9 /jr.-0 Sfi

I r ^ i + t nt*i ~ h tii> + <>£*! - -

* iCSfi*! * Sfi)AKi " 0

Writing 2 = f 1+1 -» 1) and assuming that Bi+i e Bi = B, the

desired for i the energy equation is obtained :

i i < s “ i . | 2 ~ C " ! 1 * ( T w - ’ i1 4 1 ( 8 - « 4 . t * * 0 < 3 ' 2)

The values of y^ and Q| at each point in the model obtained from this run can be stored in a file t').- use as initial condi­tions with flood hydrogreph runs, ftuyi.ded that the initial flow of the flood hydrograph corree to the steady state

3.1.5 Selection of the weighting coefficients and time step

With the weighing coefficients 0 and # set at 0,5 the discreti­zation used in the finite difference approximation of the flow equation corresponds to a centred-difference approximation and is thus of second order accuracy. However, for values of the Courant Number greater than unity parasitic oscillations appear in the solution with 6 “ 0,5 which disappear with the numerical damping in the scheme for 0> 0,66. Lig.zett and Cunge 1975 re­commend values of 6 given by 0,6 < 0 < 1,0

A disadvantage with setting the value of 8 close to unity whenusing relatively large time steps in the computation is that the errors introduced in the conservation of mass in the system become significant. With 8 = 1 and # = 0,5 the discretizedcontinuity equation can be written :

(,£] - y ^ > * B s f ’ (y»*1 - y?) 1

Ax/2= 0 (3.3)

The primary source of the error is that the net inflow to'the control volume is expressed only in terms of the flow at the end of the time step, rather than the average inflow during the time step. Similarly, the volume change is expressed only in terms of the surface width at the end of the time step, rather than the average surface width.

The effect of varying & is demonstrated further in Chapter 4 with the application of the model to two natural watercourses and a prismatic channel. The effect of varying the value of i|i was not investigated.

The (election of the size of time step to be used in the compu­tation should be baaed on obtaining a reasonable description of the inflow hydrograph at the upstream boundary or a tidal condi­tion at the downstream boundary if such exists.

The inly definitive way to see whether a time step is too large or not i,- to simulate the same event on the model using succes­sively smaller time steps. If the size of the time step signi­ficantly affects the solution, it is too large. Generally values of the time step between 0,01 and 0,25 can be used, the lower limit in situations with steeply sloping hydrographs, the upper limit for low-flow tidal simulations. It is shown in Chapter 4 that: the degree of non-linearity in the relationship bitween flow depth and =lr-ja can affect the stability of the computation. Where the flow depth to arsa relatic'iship is highly non-linear, as occurs often with natural channels, the size of the time step must be reduced to eliminate this insta-

3,2 Main complication

The flow diagram for the main computation is shown in Figure3,1. The computation proceeds with the determination of the section properties, namely the flow area, surface width, conve­yance, and momentum correction factor at all cross-sections with the initial conditions imposed. The dependent variables Q and y at each point at the end of the first time step are set equal to thf* initial values for the first iteration and the coefficients A1 to El and A2 to E2 calculated from Equations (%.7) and (2.15) and the double sweep algorithm is then applied, from which the first approximations to the dependeut variables at the end of the first time step are obtained. The section properties at the end of the first time step are then calculated using the first approximations to the dependent variables and the coefficients A1 to 81 and A2 tc E2 recalcu­lated, The double sweep routine yields the accepted values for the dependent variables at the new time step for which the section properties are then calculated. The program has the facility to print these values after every time step, or if specified by the user, only after selected cl:ne periods. The computation then proceeds to the next time step,

V I

M f

M - & Yj

Flow Chart tor Main Computation

1

1

In order to limit the memory required in the computation the values of the dependent variables and the associated section properties at the two time levels required in the computation, i.e. at time level nAt and (n+l)At, are retained in memory, all information at previous time levels being overwritten. The first step at the new time level is to transfer the values oJ the dependent variables Q and y and the associated section properties from time level (n+l)At to time level nit variables. The values of the time level (n+l)At variables are retained as initial values for the first iteration at the new time step. The procedure then repeats itself, two iterations being carried out at the new time level.

It can be seen that at each time level the section properties at e-ich cross-section are calculated twice. Initially the pro­gram was set up such that any cross-section of Np points would be divided into Np-1 sub-sections and the section properties calculated for each sub-section aiid then aummated for the cross-section. Indeed the kinetic energy and momentum correc­tion factors, a and 0 can only be calculated in this way.

Obviously where the cross-section shape is complex, with a large number of sub-sections, the calculation of the section properties in this manner becomes the most time-consuming part of the computation cycle.

An alternative method of obtaining the section properties is to interpolate them from tabulated values, the table of values being set up at the start of the run. It is only in the esti­mation of the non-linear functions, namely the area, conveyance and the momentum and kinetic energy correction factors, that inaccuracies are introduced by the interpolation. These inaccu­racies can be eliminated in the case of the area and reduced in the case of the conveyance by expressing these quantities in te. is of those which csn be directly interpolated, namely the surface width and wetted perimeter, and, in the case of the conveyance, interpolating a composite Manning roughness value for the cross-section. For example, for a water level yn+*

falling between two values yj and yj+j in the table of section properties, the area and conveyance can be calculated from

Ai 5 ” AJ + i (Bs"^+ BsjXy® ' - yj)

Where Bs 1 _ 1 and P{n+5are the interpolated values of thesurface width, Manning roughness coefficient and wetted peri­meter respectively, and Aj and Bsj are the tabulated values of the area and surface width corresponding to the water level

Since the section properties can be directly calculated in this manner the speed of computation is greatly enhanced.

3.3 Presentation of Results

The output from the model can take a number of forms. During the computation the values of Qn+* and yn+l can be printed after each time step or after a preselected number of time

A plot of the inflow and outflow hydrographs can be obtained, useful in that it presents the user with an immediate impres­sion of the degree of attenuation and the time lag in the reach. For tidal computations a plot of the water levels at both ends of the reach is also made.

Since the maximum water levels reached during the flood and the maximum flows at each point in the reach are usually also of interest these are printed together with the times at which these maxima occur. Since flow reversals can occur during tidal computations, the minimum water levels and flows are tabulated if the flow is tidal.

The results of a mass balance calculation are provided to allow a check on the integrity of the computation. The error in the mass balance calculation is generally larger for 9 = 1 than for 8 * 0,5, as has been explained in Section 3.1.4.

APfrLICATTON OF THE MODEL

4.1 Introduction

The application o£ the model to two different flow situations is presented in this chapter. In the first application, described in Section 4.2, the Sw&rtvlei Estuary is modelled under tidal conditions. The Swartv'^i Estuary is on the south-eastern Cape Coast in the vicinu.y of the coastal resort of Wilderness and comprises a lake connected to the sea by the eatuary. Tidal propagation in the estuary extends to the lake, where the maximum tidal range is of the order of 20 millimetres with a variation in the mean level between spring and neap tides of 100 millimetres. Cross-sections of the estuary have been obtained and tidal measurements made at a number points in the estuary by the National Research Institute for Oceanology (NRIO) of the CSIR. This data provides an useful check on the model.

In the second application of the model, described in Section4.3, a reach of Rietspruit is modelled under conditions of the fifty year flood. As this is an ungauged watercourse it is not possible to obtain flood flow records for comparison with the model. Ideally a reach of watercourse with two relatively close gauging stations is required to test the model under flood conditions. Cross-sections of the watercourse between the gauging stations are also required. However, it is of value to examine the behaviour of the model with inflow hydro- graphs of different shapes and steepness. The effect of the steepness of the wave front on the stability of the solution is demonstrated and the limiting effect of the non-linear stage- area relationship common to watercourses such as the Rietspruit on the size of the time step is shown by introducing the same hydrographs to a wide prismatic channel.

4.2 The Svartvlei Estuary

The Swartvlei Estuary and lake syutem is shown in figure 4.1. The take has a surface area of nine square kilometres at its average water level of +0,7 tn above Land Levelling Datum (LLD). It is largely flat bottomed, with a bed level in the deepest section of -11 m LLD. Tour rivers drain into the lake, the Diep, the Klein Wolve, the BoSkraal and the Karataia rivers which have a combined catchment area oE about 350 squsre kilo­metres. The estuary itself is some seven kilometres in length, with a surface area of approximately one square kilometre and comprises three sections. Tho upper estuary between tide gauges 3 and 4 is about 2,5 kilometres long with a main channelabout 50 m wide in a 700 m wide flood plain. The middleestuary, between tide gauges 4 and 5, is 3,5 kilometres long, but with a more confined flood plain. The lower estuary is about 1,2 kilometres long and has a dynamic character as aresult of the strong tidal flows.

Cross-section data for the estuary was obtained from the NRIO report on the hydrographic survey of the estuary (NRIO 1975) and that on the hydraulic study of the estuary (NRIO 19/8). The 47 cross-sections used are givt>n in Appendix Cl, To pre­vent the occurrence of negative water levels during thecomputation a constant of 3 metres has been added to all levels. Land Levelling Datum (LLD) thus corresponds to +3,0 metres in the model.

Manning roughness values have been selected to minimise the contribution of the areas outside the main flow channels to the conveyance of the section. This obviates the necessity for defining separate widths for the flow channel and for the flood plain, the former being used in the dynamic equation, the latter in the continuity equation. For the main flow channels the Manning roughness was varied between 0,04 and 0,06, while for the flood plain the values were varied between 0,1 and 0,3.

■ X " ' . V ' .■ ^ . : v :

' " 4

It is a requirement of the cross-sections used in the model that they provide an accurate representation of Che water surface area of the estuary, and hence tne volume in the ssKuary, at all depths of flow encountered while modelling. It can be seen front Figure 4.1 that the cross-sections are gene­rally at tight angles to the flood plain and not to the main flow channel. Consequently the length of che main channel between cva cross-sections is in some cases longer chan the distance between the cross-sections. To take account of this the roughness coefficients used in the main channel have been increased accordingly. Once the correct surface area represen­tation is obtained from the cross-sections, the selection of the roughness coefficients provides the chief calibration tool in the model.

For the upstream boundary condition tfre inflow was given as Q = 2mVg, In the application of the NRIO model to the estuary zero inflow was found to result in a net outflow from the lake while an inflow of 2 «r/s, s>».responding to the combined ave­rage annual Inflow from the four rivers to the lake, was found to result in a constant mean level in the lake.

For the downstream boundary condition in the model the ocean tide levels were used, reproduced by the simplified tidal equation given in Section 3.1.3 by Equation (3,1). The amplitude used for the semi-diurnal lunar (M2) component was 0,68 metres and for the solar component (S2) was 0,35 metres. These amplitudes correspond to those for the nearest ocean tide recording station at Mossel Say, some 40 kilometres to the west. Although the recording ptacion at Knysna is closer, this is situated on the Knysna Lagoon and therefore does not reflect the ocean tide. Mean sea level at the mouth of che Swartvlei estuary is 0,16 metres above LLD. To this musk be added an allowance for the wave set-up at the estuary month. In the calibration of the NR10 model a wave set-up of 0,13 metres was assumed, making the local mean sea level +0,29 m above LLD or, with the constant added, *3,29 m above the model datum.

For the ini - ■' conditions a constant water level of +3,75 metres and a fjr.. rf 2m3/s was used at all cross-sections. To obtain a correa;.mding level of the tide from the downstreamboundary condition the time of the tide was set to -19,33hours. The model was than subjected to a number of stabilising runs of 360 hours, approximately 30 tidal cycles. In theseruns the values of the roughness coefficients were varied forcalibration purposes.

The NRIO model was calibrated using measurements obtained during the spring tide of 15th May 1976. Since these measure­ments are presented in detail in the NR10 report (NMO 1978) they have been used to calibrate the model developed in this project.

It was found that time steps of up to 0,5 hours could be usedin the calibration runs with 8 « 1 although oscillations appearin the solution. With 0 = 1 the errors in the mass balance cal­culation for the run, in which the change in volume in the reach was compared with the net inflow to the reach over theperiod of the run, were found to be as much as 20%. The mostsatisfactory results in terms of smooth outflow hydrographs were obtained with a time step of 0,1 hours and 6 = 0,6. Compu­tational time for e 360 hour run on the HP 9845T machine was about four hours, With the section properties of the cross- section calculated individually for each sub-section and sum- mated, as described in Section 3.2, the stabilizing runs were taking 13 hours to compute, making calibration extremely time- consuming. The graphical output from a 360 hour run is given in Figure 4.2. The spring tide cycle and its effect on thelake water levels are evident in this figure.

The period of the spring tide cycle as determined by the simpli­fied tidal equation is 354,4 hours. After the stabilizing runof 360 hours the tide state in the model corresponds to 340,67 hours, since the initial tide state was -19,33 hours, Imposing a further two tidal cycles on the model after the stabilizing run carries it through the spring tide condition. The water

- UPSTREAM FLOW— DOWNSTREAM FLOW

TIME (HOURS)

TIME (HOURS)

ige and Discharge Hydrographs for ) Hour Run

levels and flows in the estuary after the 360 hour run were |therefore stored as initial conditions and a 25 hour run a’'icarried out, starting th a tide state of 340,67 hours. The jresults of this latter ruu were then compared with the measureddata for calibration purposes. The output from the model forthis run are given in Appendix C2. . ",

The tidal envelope for the estuary is given in Figure 4.3. Also ,shown are the maximum and minimum levels measured at the -various tide gauges during the calibration tide. It can be 5, !seen that the tidal wave is increasingly damped as it moves up " jthe estuary from the sea. The damping is strongest in the , flower estuary and is only slight in the middle estuary with vjlittle decrease it) the cidal range. The increased flood plain .-•[

width of the upper estuary causes further damping.

The water levels as measured at the various gauges during one complete tidal cycle are compared with those predicted by the model in Figure 4.4. The agreement between the predicted and measured data can be regarded as satisfactory.

Finally in Figure 4.5 the flows through the estuary mouth as • \

measured and predicted by the model are depicted. Again the - Iagreement is close. It was found that the flow through the jestuary mouth was more sensitive to variations in the roughness ^coefficients than was the tide levels at the various gauges. , )The maximum inflow to the estuary therefore provided the first 4indication as the whether the roughness values selected were !generally coo high or Coo low and these were then factored in •the next run. The results shown in Figures 4.2 to 4.5 and pre­sented in Appendix C2 were obtained in this way, with factors - 'of 1,5 and 3 respectively applied to the main channel and flood plain roughness coefficients given in Appendix Cl. Adjustment 6of individual cross-section roughnesses was then carried out to ^ .a very limited extent to obtain a reasonable fit between * '(measured and predicted tidal levels.

jge 6 ( 1 5

3 gh - w a te r ! --------( l 5 / 0 i i / ? 6 )

4 , 0/ 7 6 )ige 4(15 /Tide ga

Tide ga ige 4e- g f f u g e 5 ( l 5 / 05 /7 5 ) '

( 1 5 / 0 5 '76)

•Lin

(15 0 5 / 7 6 )3 , 0 -

2 , 7 -

2 ,6 -

2 , 5 -

2 , 4 -

45 4 0

Cross - sec t ion No,

I

3,0-Time (h o u rs )

Comparison of Measured and Predicted

Flows p red ic ted by model

Tim e ( h o u r s )

Figure 4.5 .Comparison ol Measured and Predicted

4.3 The Rietspruit Watercourse

The reach of the Rietspruit to which the model eras applied is located in the Brakpan Municipal area, upstrea.'i of the con­fluence with the Withokspruit. The catchment area is shown in Figure 4.6.

The area of the catchment at the head of the reach is 37 square kilometres. With two tributaries discharging into the reach, one with a catchment area of 19 square kilometres, the other with a catchment area of 6,5 square kilometres, the catchment area at the downstream end is significantly greater, being 73 square kilometres. Because of the size of the tributary catch­ment relative to the catchment at the head of the reach the reach uhould be divided into subreaches and the model applied to the channel network. However, for the purposes of this project the reach has been left undivided since the model1 can only be applied to single channel reaches. Furthermore a reasonable length of channel is obtained thereby which has only gradual variations in cross-section shape and which has a bed slope flat enough to result in subcritical flow in the reach for the range of flows applied.

Cross-sections along the reach were obtained from 1:2000 topo­graphic mapping with contours at 0,5 m intervals. The water­course consists typically of a wide flood plain with no defined channel; the definition obtained from the topographic mapping is thus acceptable. The cross-sections used are listed in Appendix D.

The Manning roughness coefficients were selected in the range 0,04 to 0,05 for the deepe?? sections of the crosa-sections, and between 0,06 and 0,08 on the flanks. The vegetation in the flood plain varies from extensive reed growth to a thick veld grass. In some areas maize is cultivated on the flood plain. The denseness of the vegetal cover to the flood plain varies seasonally; veld fires during the winter months can

R I E T S P R U iT C A T C H M E N TFigure 4.'

denude the flood plain, the vegetation only reaching maximum density at the end of the wet season. The values of theroughness coefficient used are considered representative of the watercourse in the middle of the wet season.

Three inflow hydrographs were used in the model runs, each generated by the watershed model sec up for the determination of flood lines along the Rietspruit. The representative catch­ment for the reach was assumed to be that at point A in Figure 4.6 where the catchment are< is 61 square kilometres, In the watershed model this catchment was divided into 14 subcatchments for which time-area relationships were developed. The threehydrographs were generated using the fifty year, four hourstorm over the catchment, the difference between them being in the form of the hyetograph used and the degree of attenuation introduced in the channels and two dams upstream of the reach.Hydrograph Nos. 1 and 2 were generated using a Chicago typehyetograph from which losses in the form of initial abstrac­tions, surface detention and infiltration were deducted. Hydro­graph No.3 was generated using a rectangular hyetograph with a uniform loss rate given as a fraction of the rainfall depth, equivalent to the run-off coefficient used with the Rational Method. The three hydrographs are shown in Figure 4.7. The use of the Chicago type hyetographs results in hydrographs from the individual subcatchments which have a higher peak flow andare of shorter duration than those obtained using the rectangu­lar hyetograph. For Hydrograph No. 1 routing constants wereselected so as to give what was expected to be the normal atten­uation through the water course upstream of the reach. The influence of the two dams upstream is marked, with the run-off from the tributary catchment downstream of the dams causing an isolated peak in the hydrograph. With hydrograph No. 2 minimal attenuation was introduced in the watershed, with the dams left out of the model. The resulting hydrograph has three distinct peaks. For Hydrograph No. 3 limited attenuation in the chan­nels and in the reservoirs was introduced, resulting in n hydrograph with a single peak and almost triangular shape.

Flo

w(m

3/s

)

H y d r o g r a p h N o . i

H y d r o g r a p h N o . 2

H y d r o g r a p h N o .2 4 0 -

220-2 0 0 -

180-

1 4 0 -

120-100-

8 0 -

4 0 -

20-

0 1 ,0 2 , 0 3 ,0 4 , 0 5 ,0 6 ,0 7 ,0 8 , 0 9 ,0 10,0

I- I:

T im e ( h o u r s )

> 4.7 : Inflow Hydrographs Used in Model

For the downstream boundary condition an artificial control section was introduced. This cross-section was selected so that with critical conditions occurring at the section the water levels at the adjacent section are close to the uniform flow levels for the range of flows encountered. This is equi­valent to providing a single-valued stage discharge relation­ship at the most downstream section based on the uniform flow rating curve at the section.

For initial conditions a backwater computation with a steady flow of 10 m^/g was carried out. Prior to introducing a flood hydrograph into the model it is necessary to run the model under steady state conditions for a period to allow initial perturbations to propagate out of the system. Using the results of the backwater computation as initial conditions a run was carried out using a constant inflow of 10 m^/s- With a time step of 0,01 hours and a value of 9 of 0,5, the flow stabilizes after a period of 200 time steps. The maximum deviation in the flow during the run was less than 1,4 m^/g* Using a value of 6 of 1,0 the initial perturbations were not wore rapidly damped out, the flow stabilizing after the same number of time siEeps and with the same maximum deviation in the flow.

The first run of the model was made using Hydrograph tto.l, with a time step of 0,01 hours and a value of 6 of 0,5. Parasitic oscillations appeared in the solution during the initial steeply rising phase of the hydrograph. A comparison of the inflow and outflow hydrographs is shown in Figure 4.8. The front of the wave reaches the downstream end of the reach after about 1,5 hours. At this stage the flow decreases to almost zero and then oscillates wildly, reaching a maximum of more than 360 m^/8 before converging to a smooth curve.

The outflow hydrograph from the reach obtained after the second run, in which the value of 6 was set at 1.0, is given in Figure 4.9. The dip in the outflow hydrograph is again evident after about 1,5 hours, but the oscillations apparent in the solution with 6 = 0,5 do not occur.

4

TIME (HOURS)

Outflow Hydrograph with Hydrograph No.1 and 8 « 0,5

Figure 4.8

with HydrographFigure 4.9

According to Joliffe (1984), instability in four point schemes can occur when the time step is large relative to the wave period, when the wave front approaches an abrupt form rather than a gradually varied one, or when the cross-aection has a highly non-linear relationship between the depth end cross- sectional area.

To check the effect of the aize of time step on the solution time steps in the range 0,005 hours to 0,05 hours were used in a number of runs with Hydrograph No.l as the inflow hydrogvaph. With a time step of 0,005 hours and 8 = 1 the computation stopped as a result of small depths occurring at the downstream end of the reach during the dip in the outflow hydrograph. The upper limit for the time step was found to be 0,02 hours; longer time steps resulted in unstable behaviour in the model.

Since the wave front resulting from Hydrograph No. 2 is steeper than that from Hydrograph No. 1, while that from Hydrograph No. 3 is the flattest, the effect of the steepness of the wave front on the solution can be examined. The outflow hydrographsobtained when usiny Hydrograph No,2 and 3 are shown in Figures4.10 and 4.11, The dip in the outflow hydrograph is evident with Hydrograph No. 2 but it does not occur when HydrographNo. 3 is used. It can be concluded that the dip is caused by the steepness of the wave front resulting from Hydrograph Nos, 1 and 2.

By introducing the same three hydrographs to a wide trapezoidal channel the limiting effect of the non-linear depth-arearelationship typical of natural channels on the time step used in the computation can be examined. The trapezoidal channel used is 3,41 kilometres long with a base width of 100 metres, side slopes of 1:1, a longitudinal slope of 0,1% and a Manning roughness coefficient of 0,03. A distance step of 200 metres was used in the computation.

For each hydrograph time steps in the range 0,01 hours to 0,25 hours were used and the minimum values of 8 which resulted in a

§

Figure 4.10 : Outflow Hydrograph with HydrographNo.2 and 6 = 1 , 0

DOWNSTREAM FLOW

5

TIME (HOURS)

Figure 4.11 Outflow Hydrograph with Hydro graph No.3 and 6 = 1 , 0

smooch outflow hydrograph ware determined. These values of g are given in Tallies 4.1 to 4.3 below. Where acceptably minor oscillations were evident in the solution the value of 6 is marked with an asterisk.

Time step Min value of 9 Peak Outflow Peak Outflow(h) (cumec) for 6 " 1

(cumec)0,01 0,5 157,5 156,20,05 0,6 156,4 152,00,10 0,6 155,7 149,250,25 1,0 140,0 140,0

TABLE 4.1 Minimum Values of 0 - Hydrograph No.l

Time step Min value of 6 Peak Outflow Peak Outflow(h) (comes) for 9 * 1

(cumec)0,01 0,5 189,1 187,70,05 0,6 188,0 184,10,10 0,6* 188,8 181,70,25 1,0 178,6 176,6

TABLE 4.2 Minimum values of g ~ Hydrograph Ho.2

Time step Min value of 6 Peak Outflow Peak Outflow(h) (cumec) for 9 = 1

(cumec)0,01 0,5 223,1 222,90,05 0,5 223,2 222,00,10 0,5* 224,0 220,90,25 0,67 222,9 218,2

TABLE 4.3 Minimum values of 6 - Hydrograph No.3

It is evident that much larger time steps can be used with this channel than with the natural channel. On the assumption that the solution with a ime step of 0,01 hours and 6 = 1 is the closest to the exact solution, the solutions obtained with time steps of 0,1 hours are within about 1% of the exact solution. Solutions were obtained using time steps of 0,25 hours with 6 * 1 for Hydrograph Nos. 1 and 2 and with 6 = 0,67 for Hydro- graph No. 3, although these solutions are somewhat damped. The longest time step that could be used with the natural channel was 0,02 hours. This limitation is therefore caused by the non-linearity of the depth-area relationship of the natural channel cross-sections.

The effect of the steepness of the wave front on the stability of the solution is again evident in Tables 4.1 to 4.3. Gene­rally the value of 8 required to obtain a smooth outflow hydro­graph is higher with Hydrograph Nos. i and 2 than it is with Hydrograph No. 3.

The selection of the size of the time step and the value of 6 is far more important in flood routing computations than it is in tidal computations. Where the time step, is small relative to the period of the flood wave the solution is insensitive to the value of 6 . Unless the stage-area relationship of the natural channel is almost linear, implying a well-defined channel with no flood plain flow, the use of short time steps and values of 8 between 0,75 and 1,0 is recommended for the initial runs. Lower values of 6 can be tried should subsequent runs be carried out.

5. SUMMARY AND CONCLUSION

5.1 Summary

Flood routing computacions are required to trace the changes in a flood wave as it progresses down a river channel. These changes in the shape of the flood wave can be significant where the flood wave originates in urban catchments characterized by sharp-peaked run-off hydrographs and where the amount of storage in the river channel is significant relative to the volume of the flood.

The passage of a flood wave in a river channel is a form of gradually varied flow described by the de St. Venant equations. Generally flood routing computations are based on simplified forms uf these equations, the complete equations requiring complex numerical techniques for their solution which are some­what demanding on computer resources.

The simplified methods can be classified according to which terms are left out of the de St. Venant equations. In their most simplified form the equations form the basis of the kinematic models of which the Muskingum method is a well-known and commonly used example, rhea's models incorporate artificial damping of the flood wave and the degree of attenuation is dependent on the routing parameter- o and k. These parameters must be estimated from historical flood data although in some methods they can be derived from the channel properties. Because the kinematic models ore based on a single valued rating curve, errors can occur where the rating curve exhibits a loop. In the Koussis model dynamic effects can be included to a limited extent but, as for all kinematic models, backwater effects are ignored.

The approximate dynamic and diffusion analogy models include the resistance term in the dynamic equation and can therefore reproduce backwater effects. However, because the inertia

terms are not included the effect of the variation in velocity head on the flow profile is not reproduced. These models have greater applicability than the kinematic models and can be used for most flood routing computa'ions.

The complete dynamic models, those based on the complete de St. Venant equations, can be classified according to the typu of numerical scheme used in their solution. The charvcteristics based methods are not widely used for industrial models. They do serve as a check on other methods since their solution can be brought aa olose to the exact solution as is desired and are also used to determine boundary conditions within other models. Explicit finite difference schemes are relatively simple to program. However, restrictions on the time step used in thecomputation brought about by the Courant-Lewy-Friedcrichs conditions for stability severely limits their applicability. The implicit finite difference schemes have no restriction on the time step providing a reasonable description of . .time-dependent boundary conditions is obtained. These schemes are used extensively in industrial modelling.

The finite difference scheme used in the model developed in this project is based on Verwey’s variant of the. Preissman scheme. The Preissman scheme is well documented and itsstability has been thoroughly investigated by various re­searchers. Although its formulation is extremely complex it has the advantage that only one iteration is required at each time sttp. Verwey's scheme is simpler but -squires two itera­tions per time seep to obtain a satisfactory solution. The discretized equations are written in terms of the variables at the four adjacent points (i,n), (i+l,n), (i,n*$) and (i+l.n+l). The scheme uses the double sweep algorithm for thh simultaneous solution of the unknown dependent variables at all points in the reach at the new time level. One boundary condition i.s required at each end of the reach to close the set of equa­tions. Usually an inflow hydrograph is provided at the up­stream end and a stage-time (tidal) or stage-discharge equation at the downstream end. To initiate the run the flow and water

... • ^ '

level must be defined at all points in the reach. Generally this is given by a steady state flow throughout the reach with the associated backwater curve defining the water levels.

In the model cross-section data is arranged in the form of a series of distance, ground level and Manning roughness values for each point in the cross-section. Cross-section data is stored on a file using a separate data handling program. The data is stored in a form compatible with that used in a back­water program developed by the writer. This has the advantage tba* the backwater program can be used to compute the initial water surface profile in the reach with a steady state flow that is some fraction of the peak flow of the inflow hydrograph. Reservoirs and lakes can be included in the reach, usually at the downstream end with the outlet characteristic defining the downstream boundary condition.

Inflow hydrograph data is provided in the form of a series of points on the hydrograph. Allowance has been made in the model for conversion routines to convert hydrographs generated by a hydrological model to the form used in the model.

A number of options are available in the model at present for defining the right hand boundary condition. Either a stage- discharge relationship, usually in the form of critical condi­tions, or a stage-time relationship, usually a tidal condition, can be selected.

Both the weighting coefficient 6 and the size of the time step affect the solution. For values of 8 less than 0,5 the scheme is always unstable. With 8 between 0,5 and 0,6 parasitic oscillations appear in the solution for values of the Courant Humber greater than unity. Values of 6 between 0,67 and 1,0 generally yield stable results. The value of the time step must be selected to adequately describe the time-dependent boundary conditions. Instability in the scheme can also arise when the time step is long relative to the flood wave period,

when the wave front approaches an abrupt form and when the atage-area relationship of the channel cross-section is highly non-linear. Shortening the time step in thfese circumstances will generally eliminate the instability.

In the main computation the section properties have to be calcu­lated at all points in the reach for each iteration, i.e. twice per time step. Initially the model was set up such that these values were calculated for each sub-section in a cross-section and then suramated for the cross-section. This was found t, be a very time-consuming part of the calculation and a routine was devised in which a table of values of the section properties aC discrete water levels is set up for each cross-section. The re'jjired values at intermediate water levels can then be inter­polated. This routine was found Co reduce < Hi. computation time by a factor of 3. '

The model is set up auch that the dependent variables are continuously overwritten, only Che current estimates of the unknown dependent variables at th new time level and the known values from the previous time level being retained in memory.A printout of the solution after each time step can therefore be obtained to provide a detailed description of the simula­tion. At the end of the run a plot of the inflow and outflow hydrographs can be obtained and the maximum flows and water levels at each section are tabulated. The results of an independent mass balance calculation are also printed* This is based on a comparison between the difference in the volume in the reach at the start and end of the run and the net inflow to the reach over the period of the simulation.

The model was applied to two different flow situations. In the first application the tidal motion in the Swartvlei Estuary and lake system was modelled. The estuary has been modelled by NR10 and tide gauge readings at a number of points in the estuary are available. A hydrographic survey of the estuary was carried out by NRIO and with this data, together with the tide gauge readings, a check on the model could be made. Using a constant inflow of 2 m3/s to the lake and a tidal water level defined by

Che simplified cidal equation aa the boundary condition at the sea the model was able to reproduce the measured water levels to within about 50 mm.

In this application a time step o£ 0,1 hour was used, with a value of the weighting coefficient 6 of 0,6. Errors in themass balance of about 11% were found to occur with thia value

In the second application a reach of the Rietepruit was modelled using the fifty year flood hydrograph. The Rietspruit catchment falls within the Brakpan municipal area and is gene­rally zoned for residential development. The water course com­prises a wide flood plain with no defined channel for which cross-sections were obtained from 1:2000 topographic mapping. Three hydrographs of different shapes and steepness were generated using the watershed model set up to determine thefifty year flood lines along the reach. The steepness of thewave front occurring in the channel was found to affect thestability of the computation. For the steeper hydrographs an oscillation occurs at the front of the wave which can destroy the solution. This effect limits the size of the time step to about 0,02 hours. The stability of the computation was also found to be affected by the degree of non-linearity in the depth-area relationship of the cross-sections used. By intro­ducing the same hydrographs to a wide prismatic channel, time steps up to 0,25 hours could be used in the computation without any evidence of the oscillation st the wave front. In a comparison of the peak outflow from the reach obtained fit time steps in the range of 0,01 hours to 0,25 hours with various values of 8 it was found that using a short time step with 6 ■= 1 generally yields- results which are within 1% of those ob­tained with the minimum value of 8 for which a smooth solution is obtained for the same time step. Using a longer time step the error in solution with 0 = 1 increases, values of 6 between 0,6V and 0,80 being necessary to improve the solution.

5.2 Conclusion

A flood routing model has been developed which is suitable for application to single channel reaches where subcritical condi­tions prevail. It has the advantage over more simplified techniques that routing constants do not have to be estimated, the crosa-sectiotis of the water course with estimated Manning roughness coefficients being entered in lieu thereof. Further­more, since the model outputs the maximum water level reached by the flood along the reach it is not necessary to carry out backwater computations to determine these levels, as is often required with simplified models, and no additional data chan would be used for a flood line calculation is required.

In its present form Che model could be included in a multiple channel watershed model where backwater influences between channels are negligible. Further development would permit the modelling of dendritic channel networks where inter-channel backwater effects are taken into account. In specific applications the model could be expanded to include internal boundary conditions such as weirs, bridges etc.

APPENDIX A

USER INSTRUCTIONS

APPENDIX A1 :USER INSTRUCTIONS - FLOOD ROUTING PROGRAM : FLOW MOD

The model was written on an HP9845T with 1,2 megabytes ofrandom access memory and the Structural Programming andAdvanced Programming ROMs (Read only memory). The system is connected to a Shared Resource Manager (SRM) with 64 megabytedisc storage. Data files can be read either from the SRM orfrom a local disc drive.

Setting the Default Mass Storage Device

After the program is loaded and the RUN key pressed the 1 wing prompt appears on the screen :

Local (L) or Remote (R)

The user enters (L) for a lo of the built-in tape drives drive. The prompt

Enter the local device address then appears on the screen, to ding to the type of store

For an SRM based system I iters (R), The prompt

Enter the directory paththen appears and the user enters the path to the direct: which the data files are stored or are to be stored.

When the program CURRENT MASS SI ENTER)together with displayed. If :

n the prompt (CONT if OK

entered storage devici is still applicable th

W l

presses the CONT key. To change the device address Co userfirst clears the line and Chon presses the CONT key; the program then resumes operation as described above after switch-

2. Cross-section data entry

The cross-section data file is read next. The user enters the name of the cross-ilcction data file when the prompt :Enter name of data: file containing cross-section data? appears on the screen. The data file is created using a sepa­rate program "BACIKDATA" for which the user instruction are given in Appendix A2. The program assumes that the cross- sections are ordered from the downstream end of Che reach to the upstream end, %s is required for a backwater computation.

SeCcing of Che cine-scep, Cotel i coefficient

; time and eejybting

These are encered next with che appearance of the prompc t

ENTER TIME STEP, TOTAL RUN TIME (HOURS) AND WEIGHTIMG COEFFI­CIENT. The time step and total run time are given in hours. The time step should be between 0,01 and 0,25 hours, depending on Che steepness of the inflow hydrograph and the irregularity of the channel. For normal flood routing in natural channels a time step of 0,01 hours is recommended. For prismatic channels the time step can be increased to 0,1 hours. For tidal compu­tations a time step of 0,1 hours can be used with natural channels.

The only limitation on the total run time is that it should not exceed 5 000 time steps. The inflow hydrograph data should span the total run time.

V

The weighting coefficient must be between the limits ot 0,5 and 1,0. It is desirable to have it as low aa possible so as to reduce numerical damping introduced by having it greater 0,5, However, if a small time-step is used, the numerical damping becomes insignificant and a weighting coefficient of 1,0 can ue used. A practical lower limit for the weighting coefficient is about 0,67; values lower than this often result in parasitic oscillations appearing in the solution.

4. Inflow Hydrograph Data Entry

The following options are printed on the screen for the inflow hydrograph data entry :

INFLOW HYDROGRAPH DATA READ DATA FROM FILE :1. Illudas generated2. WITWAT generated3. Generated by option 4 below4. ENTER DATA MANUALLY Select option <1, 2, 3, or 4)

Options 1 and 2 :The first two options are not as yet operative.Option 3 :Selecting option 3 causes the following prompt :

NAME OF HYDROGRAPH FILE?After the name of the file is entered the opportunity is given to factor the ordinates of the inflow hydrograph with the promptDO YOU WISH TO MULTIPLY ALL INFLOW ORDINATES BY A CONSTANT?(Y/N)To factor the ordinates of the hydrograph as read from the data file the user enters Y which causes the prompt :ENTER THE CONSTANT

If no factoring is required the user enters N to the above question. This facility is useful where the hydrograph is generated using unit hydrograph techniques. The ordinates of the unitgraph are then stored in the data file and hydrographs of various recurrence intervals can be obtained by entering a factor equal to the excess rain for the associated recurrence interval and atom duration.

Option 4 : Enter Data ManuallyWhere the inflow hydrograph data has not been entered pre­viously and stored in a file it must be entered manually. Thisis done by selecting option 4 from the above list of options.The data entered in the form of time and flow co-ordinate pairs with the following prompt appearing before each pair ia entered.

ENTER TIME (HRS) 6 INFLOW (M3/S) - (CONT WHSK FINISHED)As each pair is entered it is printed in tabular form on the screen. Once all data has been entered the user presses the CONT key leaving the entry line blank. The data can then be edited, the promptDO YOU WISH TO MAKE ANY CHANGES TO THE ABOVE DATA? (Y/tO appearing on the screen.To correct the data entered the user enters Y and the prompR ENTER NO. OF POINT TO BE CHANGED - (CONT WHEN FINISHED) appears. After the number of the point to be changed (saypoint 5) is entered the current values of the time and flow co-ordinates for point 5 are displayed in the entry linetogether with the promptTIME, INLFOW for point 5 displayed below, change as required.The user can re-enrer the co-ordinate pair entirely or., by moving the cursor to the appropriate position, change selected digits, The next point con then be edited. To exit the dataediting mode the user leaves the entry line blank and pressesthe CONT key when prompted for the number of the point to be changed.

The inflow hydrograph csh then be stored in a data file. The user is asked for the name of the file.

ENTER THE NAME OF THE FILE IN WHICH ?0U WISH TO STORE HYDRO­GRAPH DATA.The length of the file name must obviously comply with the requirements of the storage device, being a maximum of 6 characters long for local devices and 15 characters for the

5. Specifying the Downstream Boundary Conditions

Five options are available for specifying the downstream boun­dary conditions whcih are printed on the screen as follows :

RHS BOUNDARY CONDITIONS Stage-Time relationship given1. Tidal Equation (M2 and S2 components only)2. Water level at discrete time intervalsStage-Discharge relationship given3. Critical conditions4. Weir-type formula5. Rating curve co-ordinates.Select option (1, 2, 3, 4 or 5)

Only options 1 and 3 have been developed so far, these beingthe most common.

Option 1 - Tidal equation :The form of the equation and an explanation of the terms is printed on the screen as follows :

TIDAL EQUATION

LEVEL = MSL + Am% * SIN (2* P I * T/Tm2 + As2 * SIN (2 * PI *(T + Phi)/To2)

Where MSL = DATUM FOR MEAN LEVELAm2 - AMPLITUDE OF M2 COMPONENT As2 m AMPLITUDE OF S2 COMPONENT tm2 = PERIOD OF M2 COMPONENT (« 44714s)Tb2 - PERIOD OF S2 COMPONENT <“ 43200s)'i. « PHASE DIFFERENCE BETWEEN M2 and 82 COMPONENTS

ENTER Msl, Am2, As2 (all in metres) and the Phase Difference (seconds)

The four values ar6 entered, separated by commag. To achieve a constant water level as the downstream boundary condition, the amplitudes can. be entered as zeros. The hour of the tide is then entered with the prompt :

HOUR OF TIDE (0 TO 3,125 and 9,375 TO 12,5 RISING, 3,125 TO 9,375 FALLING)

By means of the ilues of phi and the hour of the tide thestate of the tide in the spring-neap-epring cycle at the startof the computation can be fixed. With both values set to zero the compuation starts with a spring tide; with Phi set at Ta2/2 and the hour of the tide set to zero, it starts with a neap tide. Similarly with Phi set to zero and the hour of the tide set to half the spring-neap-spring cycle period of 354 hours, the computation starts with a neap tide.

The values of Msl, Am2, Phi and Che hour of the bide are prin­ted on the screen. The user has the chance to correct any errors made on entering the data when the prompt DO YOU WISH TO MAKE ANY CHANGES TO THE ABOVE DATA? (Y/N).If Y is entered the user can re-enter the data for the tidal equation in ttu manner described above.

Option 3 - Critical conditions :If option 3 is selected, i.e. critical conditions specified at the downstream boundary, no further information is required to describe the boundary condition and the initial conditions can then be specified as described below.

6. Initial Conditions

There are thus two options for entering the initial conditions which are displayed on the screen as follows :

INITlot FLOWS AND LEVELS GIVEN AT EACH SECTION1, Enter datait. Read data from disc

Enter option (1 or 2)

Option 1 - Enter dataThe data is entered in the form of the flow end level at each cross-section. The program prompts the user with AT SECTION ENTER FLOW, LEVELThe value for the flow and level for the particular section are entered, separted by a comma. As the data is entered it is tabulated on the screen. After the initial data for all cross-sections has been entered the data can be edited. The editingroutine is entered by entering (Y) to the prompt :DO YOD WISH TO MAKE ANY CHANGES TO THE ABOVE BATA? (Y/N)The data for any section can be changed by entering the number of the section when prompted.ENTER THE SECTION AT WHICH EDITING REQUIRED (CONT IF FINISHED) The current values of initial flow level at the particular section are then displayed together with the prompt KD1T INITIAL FLOW, LEVEL AT SECTION ...iwtd the user can either re-enter the data line completely or change selected digits in the displayed data line using the

The editing mode can be teiroinated by leaving the entry line blank when prompted for the number of the section at which edit­ing is required.

The initial data is then stored in a data Eile for possible use in other runs. The name of the file is entered when the follo­wing prompt appears on the screen :ENTER NAME OF FILE FOB. STORING INITIAL DATA.Program operation then proceeds with the selection of the form of the output for the run.

Option 2 - Read data from discThe name of the data file containing the initial data is reques­ted with the prompt :ENTER NAME OF FILE IN WHICH INITIAL DATA IS STORED

The file containing the initial data can either be created as described under option 1 above or it can be created using the results of a backwater run. This letter method is the most convenient. The backwater program uses the cross-section data file and requires only the steady state flow in the reach to determine the steady state backwater profile. The resultsobtained using the backwater program are fairly consistent with Chose obtained from a steady state analysis by the model andthus the period required to stabilize the system prior toimposing an unsteady condition on the model is usually not significant.

7, Selection of Type of Output

There are three possible forma of printed output from themodel. Since the values of tiie flow and Isvel at all points inthe computational grid are not retained in memory the printedoutput provides the only detailed record of the run. The options are displayed tin the screen as follows :

OUTPUT TYPE1. Flow and level at all time steps2. Flow and level at selected time intervals3. Flow and level at selected cross- sections.

Enter option (1, 2 or 3)

Option 1 : Flow and Level at all time stepsWith option 1 the results of the computation at all cross- sections are printed after aeh time step. Where the time step is small relative to the total run time the amount of the print-out is excessive and it is preferable to select option

Option 2 : Flow and Level at selected time intervals The results of the computation at all cross-sections are printed after a selected time interval. The time interval is entered after the following prompt z

ENTER THE TIME INTERVAL (in hours) FOR OUTPUTThe time interval entered should obviously be a whole number of time steps.

Option 3 : Flow and level at selected cross sections This option gives the most compact form of output, The data for up to five cross-sections is output at selected time intervals. The cross-sections at which the output is required are entered with the prompt.

Enter the cross-sections at which output required (Max. 5 - eg. 1,5,8,23,44)

The time interval for output is then entered as for Option 2

8. Changing Data for a New Run

After a run is completed the user is given the facility to change part of the data and re-run the model. The options for changing the data are displayed on the screen as follows ;

CHANGE DATA FOR NEW RUN

Cross-sectionsLHS Boundary conditions

3. RHS boundary conditions4. Initial conditions5. Theta6. . NO CHANGES - END OFF.

SELECT OPTION (1, 2, 3, 4, 5 or

The program loops back to the list of options after each datachange, allowing more than one set of data to be changed, untilthe entry line is left blank and the CONT button depressed, in which case the new run is started, or option 6 is selected. If option 6 is selected program execution is terminated. If the entry line ia left blank and the CONT button is depressed the type of output required is requested as is described in Section 7 above. The remaining options are described below.

Option 1 : Cross-section dataThe new cross-section data is entered as described in Section 2

Option 2 : LHS Boundary Condition!!The new time step and total run time are first entered. Theseare requested by promptENTER TIME STEP, TOTAL RUN TIME (HOURS)Both quantities should be given in hours, The inflow hydro­graph data is then requested as is described in Section 4.

i » ";;

Option 3 : RHS Boundary ConditionsThe new downstream boundary condition is entered as described in Section 5.

Option 4 : Initial ConditionsThe user has the facility to use the state oE the reach at the end of the previous run as initial conditions (or the new run:DO YOU WANT TO USE DATA FROM PREVIOUS RUN AS INITIAL CONDITIONS? (Y/N)If Y is input the program requests the name of the file in which the initial conditions are to be stored. If N is input the new initial conditions are input as described in Section 6.

Option 5 : ThetaThu new value of the weighting coefficient Theta is requested with the prompt :ENTER VAim FOR THETA BETAKEN 0,5 and 1,0

USER INSTRUCTIONS - CROSS-SECTION DATA INPUT PROGRAM : BACKDATA

This program creates a data file containing the crose-sections o£ a river reach required for use in the one-demensional flow model PLOW _M0D or in the backwater program BACKWATER. The file name is made up of the prefix "Reac" and the number of the reach, which can be between 0 and 99 inclusive.

The program begins with the setting of b'^ default mass storage

After the program is loaded and the RUN key pressed the follo­wing prompt appears on the screen :

Local (L) or Remote (R)

The user enters (L) for a local mass storage unit such as one of the built-in tape drives on the HP9845 or a floppy disc drive. The prompt

Enter the local device address (e.g. :H7,U,1)then appears on the screen, to which the user can respond accor­ding to the type of storage device.

For an SBM based system the user enters (R). The prompt

Enter the directory paththen appears and the user enters the path to the directory in which the data files are stored or are to be stored.

When the program is re-run the promptCURRENT MASS STORAGE IS (CONT if OKAY - LEAVE BLANK TO RE­ENTER)together with the last entered storage device address is displayed. If the address is still applicable the user simply presses the CPNT key. To change the device address to user

first clears the line and then presses the COOT key; the program then resumes operation as described above after switcb-

The number of the reach is entered when the following prompt is displayed on the screen :

Enter the Rear1- number - (COOT IF FINISHED)

The program checks whether a data file for the reach number en­tered already exists at the default mass storage address and reads the file into memory if it does. The main menu for the program is then disr! ' on the screen as follows :

DATA FOR REACH ..1. ENTER CROSS SBC.— DATA2. PRINT BARD COPY OF CROSS SECTION DATA3. EDIT CROSS SECTION DATA4. HON BACKWATER PROGRAM WITH DATA FOR REACH ...5. ENTER A NEW REACHENTER OPTION (1, 2, 3, 4 or 5>

Option 1 ! Enter cross section dataThe data for each point in the cross-section is entered with the prompt ;ENTER DISTANCE, LEVEL, MANNING N - (COOT WREN FINISHED WITH THIS SECTION)

The distance given is that from some arbitrary origin in the cross-section. The three quantities entered must be separated by commas. As the data for each point is entered, it is dis­played in tabular form on the screen. Once all the points in the cross-section have been entered, the user leavec the entry line blank and depresses the COOT The user then has thefacility to edit the data just entered ;

DO YOU WISH TO MAKE ANY CHANGES TO THE ABOVE DATA? (Y/N)

Entering Y to this prompt for the croas-section just change data £ot other croi entered by selecting Option

illows the user to change the data entered. It is only possible to s-sectiona once all data has been 3 Erom the main menu. The editing

of data is discussed further under this opti<

4 '

Once the changes to cross-section have been made the next cross-section can be entered. The distance between the laat entered croas-section and the cross-section about to be entered is requested with the promptCHANNEL LENGTH BETWEEN SECTION ... AND ... (CONT IF ... IS LAST SECTION)If the last cross-section entered is the final cross-section in the reach the user leaves the entry line blank and presses the COHS key. The date Ear the reach is then stored by ehe program and the main menu is displayed. If a distance is entered in re­sponse to the above prompt the data at each point in the cross- section is then entered as described above.

Option 2 i Print hard copy of cross-section data A print ouK of the cross-section data for the reach on the ther*»inal printer is obtained if this option is selected.

Option 3 : Edit cross-section dataThree options are available for editing the cross-section data for a reach which are displayed as follows i EDmt$G 0? DATA FOR REACH ...1. INS5RT A NEW CROSS SECTION2. CHANGE EXISTING DATA3. DELETE A CROSS SECTION

ENTER OPTION (1, 2 or 3) - CONT WHEN FINISHED

Edit option 1 : Insert a new cross-sectionThe number of the cross-section immediately below the one to be inserted is first entered. The prompt is :ENTER NO. OF CROSS SECTION BKLOW NEW SECTION

The distance from the adjacent lower-numbered section and the data for the cro^a-section are then entered as described under Option 1 above.It should t«! nor id that the distance between the section inser­ted and the adjacent higher-numbered section retains the value of the distance interval into which the section was inserted, the length of the reach thus increases by the distance between the inserted section and its adjacent lower-numbered section.

Edit Option 2 : Change existing dataThe user first enters the number of the cross-sectin to be changed:WHICH CROSS-SECTION 00 YOU WISH TO CHANGE? (COST WHEN FINISHED) After a cross-section has been changed this prompt is again displayed to allow any number of cross-sections to be altered. Once all changes have been made the user leaves the entry line blank and presses the COHT key. The program then reverts to the Rfliting Menu.

When the user has entered the number of the cross-section to be changed the following prompt is displayed :DO YOU WANT TO CHANGE THE WHOLE SECTION? (t/N)

Entering Y in response to this prompt allows the user to com­pletely redefine the cross-section. Data entry for the cross- section then follows in the manner described under Option 1 above. Entering ft in response to the above prompt allows the user M change individual points in the cross-section. The distance between the section and the adjacent lower-numbered section con bo changed when the following prompt is displayed : CHANGE Tflti DISTANCE BETWEEN SECTION ... and SECTION ...The current value of the distance is displayed in the data enLry line. If no change j,s required the user merely presses the COST button; otherwise the new distance is entered. The next prompt is :WHICH POINT DO TOO WISH TO CHANGE IN CROSS-SECTION ... (COST WHEN FINISHED)

When che number of the point to be changed is entered the current values of the distance, level and Manning toughness coefficient for the point are displayed in the data entry line together with the promptCHARGE DISTANCE, LEVEL OR MANNING N AT POINT ...The user can then change the values as required, The data for the cross-section is then printed on the screen and the next point to be changed can then be entered. If the entry line is left blank and the COST button depressed when the prompt for Che next point to he changed in the cross-section is displayed the next cross-section can be changed.

Edit Option 3 : Delete a cross-sectionThe number of the cross-section to be deleted is requested with the prompt :WHICH CROSS-SECTION DO YOU WISH TO DELETE? (CONT WHEN FINISHED) Leaving the entry line blank causes program operation to resume with the menu for the editing options.

It should be noted that deleting a croas-section results in the renumbering of all higher numbered cross-sections. If a number of cross-sections are to be deleted they should be deleted in order of decreasing magnitude.

When no further editing is required the user exits edit mode by leaving the entry line for the editing menu blank. The program then stores the data. If a data file under the reach number already exists the user is given the option to change the reach number or purge the existing file :ENTER A NEW REACH NO, OR PURGE FILE Reac ...The file must be purged manually by typing in PURGE "Reac X"Where X is the reach number, and pressing the EXECUTE key. Pressing the CONT key thereafter will result in the new data file being stored under this name.

Option 4 : Run backwater program with data for Reach ... Selecting this option results the backwater program "BACKWATER" being loaded.

Option 5 : Enter a new reachThia option allows data for a new reach to entered. Program operation resumes with the number of the new reach being entered.

109-

APPENDIX B

PROGRAM LISTINGS

APPE.1DIX B1 PROGRAM LISTING - FLOW_MOD"FLOW MOD" I ONE DIM. FINITE DIFFERENCE

OPEN CHANNEL UNSTEADY FLOW MODELBASED ON de St Venant EQUATIONS AND SOLVED USINGTHE PREISSMAN SCHEME AS MODIFIED BY VERWEY

DIMENSION ARRAYS I

OPTION BASE 1COM Distance!100,20),Np(ISO),Leveil<20>,Mano<2®),PathtC503,Deviee#C201,Ru

DIM Reach!100,20,3).Points!160),Lx<100)Section Properties:DIM Areal!100),Area2!100),Area!10B),Bsurfl!100),Bsurf2!100),Alpha2!100),B

100)DIM BetalC100),BetaS!100),Dx!100),X(100),Ybot!100),Gg<180,180)DIM Level(100,20),B<100,20),Ar<100,20),P<100,20), Alpha!100,20), Beta!100,2

e)lPointer<26>,Mn(100,20),Courit6P<ie0)DIM Nnl!100)

Boundary and Initial Conditions:DIM Q 1hs!5000),Q rhs!5000),Q discrete!5000),Stage(100),St age out!100)DIM Qstart!100),Ystart!100),Y~1hs!5000),Y rhs<5000)

Coefficients:DIM A2!100),Bill00),82!100)DIM C2<100>,Dl<100>,D2<100),El!i80),E2!l00),F!100),G!100),H!100),I!100>DIM J<100>>K<100),Kl(100),K2!l08),Paral!160),Para2<108),Para3<160)

Main Variables:DIM Q1!100)IQ2!100),Y1<100),Y2(100)SQraax!100,2),Ymax!100,2),Qnin!100),Yfni

n<iae>Ouput control'DIM Cress„print!20)

DIMENSION STRINGS

DIM Edit$E803

SET CONSTANTS AND DEFAULTS :PRINTER IS 16

Theta=.5

Comp!ete=l

MAIN CONTROL ROUTINE***************

GOSUB Storage device ! SETS MASS STORAGE DEVICEGOSUB Data input 1 DATA INPUT MASTER ROUTINELOO0

GOSUB Main calc I CALCULATION MASTER ROUTINEGOSUB Data change i ALLOWS CHANGES TO DATA FOR NEXT RUN

EXIT IF Op ch$«"5" 1 EXITS IF NO FURTHER RUNS REQUIREDGOSUB Output type I SETS PRINT INTERVAL FOR NEW RUN

END LOOP#************«*#*****##*****##**#######***#** .-»#»#*»*»»»»»»»•»#*•*»*****•

i DATA INPUT Level II ************************************************************************* Data_input: I

GOSUB Cross sectGOSUB ReditT ! REDIMENSIONS ARRAYSINPUT "ENTER TIME STEP , TOTAL RUN TIME (HOURS),Weighting coefficient THE

Dt1,Tmax.Theta Bt“DtI*3690GOSUB Lh_bound I SETS LHS BOUND. CONDITIONSGOSUB Rh_boUhd I SETS RHS BOUND. CONDITIONSGOSUB Ini t_cond ! SETS INITIAL CONDITIONSGOSUB Output type I SETS INTERVAL FOR PRINT OUTRETURN

! *************$?*-****&.'**************************************************! MAIN CALCULATION Level 1I ************************* .,******%4,**************************************

CALCS INITIAL STORAGE FOR MASS BALANCE

INITIALIZES VALUES USING PREVIOUS TIME STEP

Run=Run+t MAT Q2=Gst art MAT Y2=Ystart MAT Ymax«ZER MAT Qnxax=2ER MAT Ymi n=(999>MAT Qmi n=<999>

GOSUB Printhead GOSUB Print

FOR I-I TO Nsections GOSUB Section prop

N_iterat ions=2 Qsum_i n=9start <I)/2 Qsum_out=Ostart <11)/2 GOSUB Storage Voi_start=Vo1ume

EXIT IF N>Nn-l MAT Y$=Y2 MAT 01=02 MAT Areal=flrea2 MAT K1-K2 MAT Betal=Beta2 MAT Bsurfl=Bsurf2 FOR Iter-1 TO N_iterations

GOSUB Ceeff i l j e.nts GOSUB Sweep FOR 1 = 1 TO i' eci ions

GOSUB Sect I on prop NEXT I

NEXT IterIF FRACT(N*Dt/Print int>=0 THEN GOSUB Print IF N>1 THEN

esuia_in=Qsum_in+Q__lhs(N)Qsura_ou t -Gsuin^eu t +G_rhs (N >

END LOOPQsuen^i n=<Qsum_i n+Q_1 hs<Nn)/2)*Dt Qsua''ciut = <U8um out+Q_rhs(Nn)/2)*Dt GOSUB Print maxCALL PI ot_hydro<Nn, Bt 1, 0 Ihs<*>, C!_rhs<*), "FLOti (CUMEC)’’)IF Rh oc=2 THEN CALL Plot hydro(Nn,Dt1,Y_1hs<*>,Y rhs<*>,"LEVEL

11 TO GOSUB Storage I CALCS FINAL STORAGE IN REACH

1240 Data_change:

PRINT PAGE," CHANCE DRTfl FOR NEW RUH",LIN(1

______ 3. RHS Boundary Condi11PR INT " 4. Initla'i Condi' ‘ - "PRINT " 5. Thet«",LIN(l)PRINT " 6. NO CHANGES - L. . ...LINPUT "SELECT OPTION <1,2,3,4,5

EXIT IF <0p_ch*="111 OR <Or,_ehS="6“) Op_c h=VRL'' 0(>_e h*)SELECT Op_ch

CONT TO START RUN",Op_c

TOTAL RUN TIME (HOURS)",Dt1,Tmax

NS ?

INPUT "ENTER VALUE FOR THETfi BETWEEN 0,5 AND

1740 | *************************************************************************PRINT PAGE," CROSS SECTION DATA".LINO)

data ?",FUe*

READ #l;Cum_np,Nxs Nseet ions=NxsREDIM Reach<Nxs,20,3>,D1stAnc

READ #i;Np(*>,Dx<*),Reaeh<*>

•<Nxs,20),Np<Nxs>,Npl(Nxs),Dx<0:Nxs-l>,C(

MAT SEARCH Np,MAX?Mp

(Nxs,Mp),Mn<Nxs,M|a),flr<Nxs,Mp>,B(Nxs,Mp),P(Nxs,Mp),fltpha<Nxs,H1870 REDIM Le P>iBeta<Nxs,Mp>

FOR Nx=l TO Nxs Z"Nxs-Hx*l Lx(I>=Dx<Nx-l>REDIM Level 1 (Np<Nx> >, Po 1 nteMNi: FOR J=1 TO Np(Nx)

D#«t*ncff<I,J>=ReaCh(Nx,J,1> Levell<J)«Reach<Nx,J,2)Manri( J >=Reach<Nx, J, 3)

NEXT JMAT SORT Level 1 TO Painter Jj-1FOR 11-1 TO Np(Nx)

J=Pointer<Ii >Msl=Leve11<J)IF 11-1 THEN Ws1=Leve11<J>+.

Beta<I, Jj)>'

3230

EXIT IF I(+E>Np<Mx)EXIT IF Level 1(J><>Level1<Pointe

END LOOP 11=11*E-1 CALL Secti n_tab1e<I,Nx,l''si,B<I, Jj>,Ar<Z,Jj>,K,P<I, Jj>,A1pha<I,JJ)

Level<I,Jj >*Leue11<J)Mn<11JJ >=Rr<I,Jj)*<Ar<I,Jj>/P<I,)A<2/3)/K Jj=Jj+l

NEXT Ii NpK I W J - 1

NEXT Nx •MAT Np=Npl MAT Dx=Lx Ji=Nsect1ons Xx=Nsect1bns RETURN

, REDIMENSION ARRAYS Wei 22240 I ************************************************************ ************2250 Redim: i2260 REDIM Areal<Nxs),Area2<Nxs),Area<Nxs>,Bsurfl(Nxs>,Bsur-f2<Nx8),tilpha2<Nxs) ,SsurP<Nx5)2270 REDIM Betal<Nxs),Beta2<Nxs>,Dx<Nxs),X(Nxs),Ybot<Nxs>2280 REDIM A2<Nxs>,B1<Nxs),B2(Nxs)2290 REDIM C2<Nx»>,T1<Nxs>,D2<NX8>,EI<Nxs>,E2<Nxs>,F<Nxs>,G<Nxs>,H<Nxs>,I<Nxs>2300 REDIM J(Nxs),X<Mxs),K1<Nxs>,K2<Nxs),Paral<Nxs>,Para2<Nxs>,Para3<Nxs)2310 REDIM Qmax<HxS,2>,Ymax<Nxj,2>232023302340

RETURNI *************************************************************************i LHS BOUNDARY CONDITIONS Level 2I ************************************************************************* Lh bound: I

PRINT PAGE,- INFLOW HYDROGRAPH DATA: 11. LIN<2>PRINT " READ DATA FROM FILE I",LINO)PRINT 11 1. IHudas generated"PRINT " 2. W1TWAT generated"PRINT * 3. Generated by option 4 below",LIN(l)PRINT " 4. ENTER DATA MANUALLY",LINO)INPUT " Select option < 1,2,3 or 4>",0p_inflo SELECT Op_i nf1o

GOSUB Hydro_read

GOSUB Hydro_1nputGOSUB Hydro store

1 CASE ELSE

INPUT "Enter 1,2,3 OR 4 TO INDICATE OPTION I!",Op InfloGOTO 2440

END SELECT -114-GOSUB D i scret »_hydro MAT i3_1 hs=Q_discrete REDIM Q_rhs<Nn)

RETURN

I RHS BOUNDARY CONDITIONS Level 2I ************************************************************************Rh bound: I

PRINT PAGE," RHS BOUNDARY CONDITIONS \LIN<2)PRINT " Stage-Time relationship given",LIN<1)PRINT “ 1. Tidal Equation (M2 AND S2 components only>",LIN<l>PRINT * 2. Water level at discrete time intervals",LIN<I>PRINT ” Stage-Dlschange relatlonship given",LIN(1>PRINT " 3, Critical conditions",LINC1>PRINT " 4. Weir-type formula",LIN<1>PRINT " 5. Rating curve co-ordinates",LIN(I)INPUT " Select option <1,2,3,4 or S)",Rh_bc SELECT Rh_bc

PPINT PAGE;" TIDAL EQUATION :",LIN(2>PRINT "LEVEL” MSL+ Am2*SIN<2*PI*TVTM2+Rs2*SIN<<2*PI*T+Phi>/Ts2>",LIN<2

MSL=DATUM FOR MEAN LEVEL"Rm2=AMPLITUDE OF M2 COMPONENT"As2=AMPLITUDE OF S2 COMPONENT"Tir2=PERI0D OF M2 COMPONENT .<=44714 s>" Ts2=PERI0D OF S2 COMPONENT (=43200 s>" Phi-PHRSE DIFFERENCE BETWEEN M2 AND 82 COMPON

>,As2 (metres), and the Phase Differed

PRINT " w

INPUT "ENTER MSL <m),Am2 (metn .econds)“,Msl,Am2,As2,Phi

PRINT " M8L= Msl;" m"PRINT " Am2= ";Am2;" m"PRINT " As2« -y.fRsE; " m"PRINT " Phi = ";Phl;" s"INPUT "HOUR OF THE TIDE <0 TO 3.125 and 9.375 TO 12.5 RISING ,3.125 T0

5 FALLING)",Tide statePRINT LIN(1)," Time of Tide- *jTide_state;" h"GOSUB EditIF EditSCl,l]=“Y" THEN 2840Ttn2=44714Ts£=43200Tlde_stateeT1de_state*3600

CASE ELSE

D1SP " INCORRECT ENTRY - TRY AGAIN - ";GOTO 2740

END SELECT RETURN

I ************************************************************************I INITIAL CONDITIONS Level 2I ************************************************************************

Init_cond: IPRINT PAGE," INITIAL CONDITIONS :".LIN<2>PRINT PAGE," INITIAL FLOWS AND LEVELS GIVEN AT EACH SECTION".LIN<2)PRINT " 1. Enter data",LIN<t>PRINT « 2. Read data from dise",LIN<l)INPUT "Enter option <1 or 2)",Op t n l t S SELECT 0p_init2

PRINT PAGE," INPUT OF INITIAL FLOWS AND LEVELS".LIN<2)FOR 1=1 TO Nsect1ons

DI3P "AT SECTION I;INPUT " ENTER FLOW,LEVEL",Qstart<I>,Ystart<I>PRINT USING " 3X,4D,5X,4D,DD, 5N, 4D,3D"51,Qatart <I>,Ystart <I>

GOSUB Edit

. ’r

D

3220 IP E d i m i . n — Y» THEN _115_3230 LOOP3240 LINPUT “ENTER THE SECTION AT WHICH EDITTING REQUIRED < CONT IFFINISHED ) Ed it$

3256 EXIT IF EdU#-""3260 I=VAL<Ed1t»>3270 Edit#=" M6,VAL#<«8t4rt<I>>&Comnia*8.VAL$<Y6tar’t<I>)3260 DISP "EDIT INITIAL PLOW .LEVEL AT SECTION "jI»3290 EDIT Edit*3300 COSUB Strings3310 Qstart<n-VAL<Str#<l>>3320 Ystart <n=VAL<Str»<2)>3330 PRINT USING " 3X,40,5X,4D.nn,5X,40.38";I,Cstart<r>,Vstart<I>3340 END LOOP3350 END IF3360 GOSUB Ihit_store3370 CASE 2336® GOSUB I n i t -cAd3390 CASE. ELSE3460 BEEP3410 INPUT "ENTER i OR 2 TO INDICATE OPTION II”,Op 1nft23420 GOTO 31303430 END SELECT3440 Q lha<l)=Qstart<l)3450 Q (-hs<l)-Qstart<Xx)3460 Y 1hs<l)-Ystart<l>3470 Y“rhs<l>=Ystart(Xx)3480 RETURN3560 I ************************************************************************ 35}0 ! OUTPUT TYPE Level 23520 I ***********************************************************************3530 Output type: I3540 PRINT PAGE," OUTPUT TYPE : ",LIN<2>3550 PRINT " 1. Flow and level at all Time Steps",LIN<1>J1S0 PRINT " 2, Flow and 3eve! at Selected Time Interval*,LIN<1>9570 PRINT " 3. Flow and level at Selected Cross^sect1ons",LIN<t)3580 INPUT “Enter option <1, 2 or 3)",0p_out3590 SELECT Op out3600 CASE 13610 PMnt_int = l3620 CASE 23630 INPUT "ENTER THE TIME INTERVAL (In hours) FOR OUTPUT “,PMnt_int3640 Pf'1nt_int=Pr1nt fnt*36003650 CASE 33660 LINPUT "ENTER THE CROSS SECTIONS AT WHICH OUTPUT REQUIRED (MAX 5 - eg.1,5,0,23,44",Edit*

3679 GOSUB Strings3600 FOR 1=1 TO A!13696 Cross p r t M <I>=VAL<Str*<I>>3760 NtiXT I3710 INPUT "ENTER THE TIME INTERVAL <in hours) FOR OUTPUT u,Print_if,t5720 Print infPrlnt lnt*36003730 CASE ELSES740 BEEP3750 DISP " ERROR ON DATA ENTRY -3760 GOTO 35803770 END SELECT 3780 RETURN3790 | ***********************************************************************3900 I CALCULATION OF SECTION PROPERTIES Level 23910 | ************************************************************************3620 Section_propZ I3830 Step*!3840 IF Y a c m L e v e K I . l ) THEN Y2<I)=Y1(I)3950 Wd=Y2<n-Level <1, 1)3860 IF N»t THEN 3870 Ccunter<I)=l3860 ELSE3990 IF Y2<IXY1(I) THEN Step— 11960 END IF

453645464550

LOOP -116-EXIT IF Y2<I)>=Leuel<I,Np<I)>EXIT IF <V2<I>>*Ltvel<IlCounter(I)>) AND (Y2< I XLeuel(I,Counter<I) +1)>

IF Counter<I> >1 THENIF Y2(I)<L*va1(I,Count*f(I>> THEN St«p»-1

IF Steps-1 THEN Step«l

Counter< I >==Countef‘< I>+Step END LOOPIF Y2< I XLeve 1 <I,Np<I>> THEN

J=Ceunter<I>Dy=Y2<I)-Leye1 <1, J>F-Dy>'<Leve1 <!, J+$)-LeveKI, J)>Db=<B<I,J+l>-B<!,J>)*F Bsurf2< I)=B<I,J)+DbArea2<I>=Ah<I,J)+<Baurf2<I>-Db/2>*DyPerii«=PU, J+I)-P<I, J))*FMan=Mn<I, JH<Mn(I, J-H)-Mri<I, J))*FK£< 1 >=Area2< i) *<Fire&2<l>',Per-in>''(3/3>/Mslr>fl1ph*2<I>rfi1ph*<I, J>*<A1phe<l,.J*l>-Fllph*.<I,J>>«FBets2< I>i»Beta<IlJ> + <Beta<I,J+l) -BetaC I, J>) *F

Dy=Y2<I>-Levei(I,^F»Dy/(Leve3 (I, J)-LewKI, J-D)Bsurf2<l)=B<I,J>Area2< I >=Ar<I,J>+Bcurf2<I>#Dy Perl ia=P<I, J>M*n=Mn<I, J>K2< I >=Area2< I ,V*< Area2< I >>,Per1 m> A<2>,3>/'Mari Alpha2<I>=A1pha<I,J>Beta2<I>=Bet a< Z,J)Ceunter<I)=Np<I>

RETURNI **********************************************************************I CALCULATION OF VOLUME OF STORAGE IN REACH Level 2| **********************************************************************Storage; I

FOR 1=1 TO Nsectlons-lVot uifieaVo1 Ume*<ftrea2< I )*Area2( 1 + 1 > >/'2*BxC I >

I **********************************************************************I CALCULATION OF COEFFICIENTS Level 2I ************************************************************************Coefficients: I

fll=-ThetaCl«ThetaFOR I=t TO NaectioKS

K<I)=Kt <I>*<1-Theta>+K2<I)*Thet&B»urf<I>=Bsurfl<I>*<l-Theta>i-B5urf2<I>#Theta Area<I> = <1-Theta)*Area$ <I)*Theta*Area2<I>Paral<I> = < <1-Theta>*Bet at <I>+Theta*Beta2<I>)*QI<I>/’ftrea<I) Para3<I>=9.ai*ABS<ai(I)V<KKI)*K2<I))

FOR 1=1 TO Nsectlons-l Dx=Dx<I)AreaaPsi *Area( 1 + 1 ) + <i-Psi>* Ahead) Para2<I>=9.8i*<Ps1*Area<I*l>+<t-Ps1)*Rre

Bl<I>=Bsurf<I> *Dx/Dt*<I-Psi) ni<I)=Bsurf<I+l>*Dx/Dt*PslEl(I)=(l-Theta)*QI(I>4'Bl<I)*Yl<I>-<l-Theta>*Ql<I+l>+Dl(I)*Yl(I+l>

<

I INERTIA TERMS OMITTEDIF Comp)

COMPLETE DYNAMIC EQUATION

E2<I)=-Dx/Dt*<<l-Ps1>*Ql<n*Psl*Qi<I + l>>+Para2<I>«<l-Theta>*<Yl<I)-Y

Left Hand Boundary Condi11 eh

Stage gi

I(I>=-Dl<I)/GamgaJ-< !>■<£!< I >-fti*G<I>>^GammaKappa»ftIfa*H <I> +02 <I)F<I*I>=-<A1fa*I(I)+D2<I>>/KappaG<I+l>=<E2<I>-S2<I)*G<I>-Alfa*J<I>>/Kappa

SELECT Rh_bc I Right Hand Boundary CondHI;ASE t I Tidal equation

Y2m>=St&ge„out<N>

Froude=A1pha2<Ii)*Q2<Ii>A2*Bsurr2<H)/<y.ei*Ai'ea2(

Dy=< 1-Froude) A2*Area2< I i VBsurf 2< I i >*8GN< l.-Froudi IF fl$S<Dy>>Ud/2 THEN Dy=Wd/2^" ' ->Y2<!i>=Y2<H)-Dy GOSUB Sect Ion_propl

■ n - .

5320 Y2<I)=H(I)*Q2(I+1>+I(I)*Y2(I+1)+J(I)5330 Q2(I)-F<I)*Y2<i)+G<I)5340 END IF5350 IF IterM THEN5366 IF Y2<I)>Ymex(I,8) THEN5370 Ymax<I,I> = N*Dt i5360 Ymax<1,2 >=Y2<I>5390 END IF5400 IF Q2<I)>Qmax<I,2> THEN5410 Q«ax< I 1 >=N*Dt 15420 Omax<I,2>=Q2<I)5430 END IF5440 IF Y2<I'<Ymfn<I> THEN Ymfn<I)=Y2<I)5450 IF Q2<IXQmirt<I> THEN Qml n< I )=Q2< I)5460 END IF5470 NEXT I5480 IF IterM THEN5490 0_rhs<N+l>=Q2<Xx.5500 Y_1hs<N+l)=Y2<l)5530 Y rhs(N*l>»Y2<Xx>5520 END IF 5530 RETURN5540 I ************************************************************************5550 I PRINT HEADING Level 255(50 ! ************************************************************************5570 Printhead: !5560 PRINTER IS 05590 PRINT PRGE5660 Lines=Q5610 PRINT "______________________________________________________________________

5620 PRIMT LIN(I>,"5630 IF CO#pi6te=0 THEN 5640 PRINT "5650 ELSE 5660 PRINT "5670 END IF5680 PRINT “____________

FINITE DIFFERENCE FLOW ANALYSIS"

using the APPROXIMATE DYNAMIC EQUATIONS 1

using the COMPLETE DYNAMIC EQUATIONS "

5690 PRINT " DATA FILES :“5700 PRINT " Cross sections : ";File*5710 PRINT " LHS Boundary Conditions: "j5720 SELECT Lh be 5730 CASE t5740 PRINT "Inflow hydrograph: "JInflo*5750 END SELECT5760 PRINT “ RHS Boundary Condition®; *$5770 SELECT Rh be 5760 CASE 15790 PRINT "Outflow hydrograph: "jOuiflo*5606 CASE 25610 IF <A»2a0> AND <As2=0> THEN5820 PRINT "Constant water level! '"Ms)5830 ELSE5840 PRINT "Tidal variation: ‘5850 PRINT USING ‘‘27X, 18A, DD, 2D,2A"; "Mean sea level : ";Msl; " m"5660 PRINT USING "27X,ISA,DD.3D,2A";"M2 amplitude : ";Hm2;" m"5870 PRINT USING "27X, ISA, DD. D6,2A" i "S2 aapMiude : ";As2;" m"5860 PRINT USING "27X,ISA,5D,5A";"Phase Difference: ";Phi;" s"5890 PRINT USING "27X,t7A,3D.2D,2A,/ll;"Tiaie of Tide : "; Ti de_state/3600," h"5900 END IF5910 CASE 35920 PRINT "Critical Flow Conditions"5930 END SELECT5940 PRINT " Initial Conditions ; ";Inlt*5950 PRINT LIN(l);" THETA : ";Theta;" PSI : ";Psi5960 PRINT USING " ISA, 2. DD, X, 5A"; " TIME STEP : Dt/3600; "hours1'5970 PRINT ",_____________________________________________________________________ __

5960 Prl . page: I HEADING MT TOP OF PAGE FOR SELECTEDI CROSS SECTIONS

IF 0p_out=3 THENPRINT USING "/,6m,DD,m,/ ":"RUN “:Run:""PRINT USING "#,5X"FOR 1=1 TO fil1

PRINT USING 3X,$A,DD"i"SECTION",Cross printCI)NEXT I PRINTPRINT USING 8A";"TIME"FOR 1 = 1 TO AT 1

PRINT USING "#.2X.7m.2X.SA.X":"STAGE","FLOW"

PRINT USING

RETURN6140 I ** * * * * * * * * * * * * * * * * * * * * » * * * * * * * + * * * * * * * * * * * * * * * * * * * * * * * * * * * •* * * * * * * * * * * + *61S0 I PRINT OUTPUT Level 26160 I **************************************9********************************* 6170 Printi I6180 PRINTER IS 0 6190 SELECT Op out 6200 CASE 1,2 6210 PRINT USING 1

TIME ":N*Dt/3600.'SECTION

- ;i,V2a>,Q2<i>

LE6226 PRINT “ SECTION LEVEL FLOWVSL FLOW ".6290 IJ=INTkNsections/2)6240 FOR 1=1 TO Ij62S0 PRINT USING “2<2X,5H,5X,SD.3D,X,M6D.2D,4SO ";I,Y2<I>,02(1>, H I j ,Y2<I+ 1 JiV > Q2< I + IJ >6260 NEXT I6270 IF FRfiCT<Nsections/2><>0 THEN6269 I“Nse«t1ons6290 PRINT USING ”38X,5D,SX,5D.3D,X,tv6300 END IF6310 CASE 36320 PRINT USING 3D.2D";N*Dt16330 FOR Ij=l TO fil16340 I “Cross prlnUIJ)63S0 PRINT USING X,DD.3D,4D.2D"5Y2<I>,Q2<I>6360 NEXT Ij6370 PRINT6369 IF FRflCT< «',es-40>/62)=0 THEN6390 PRINT r6480 GOSUB P <n page6410 END IF6420 LInes=L1nes+l6430* END SELECT6440# PRINTER IS 166450' RETURN6460 I *******************************4*****','-*******************************6470 I PRINT MAXIMUM STAGE / FI. -I! VALUES .Level 36460 I MINIMA FOR TIDAL F L O W ONLY6490 I *************************************- f********************************6500 Print max: I6510 PRINTER IS 06520 PRINT USING __________________ ____________________________________

PRINT USING "K,//' PRINT USING "*,K"i IF Rh bc=2 THEN

PRINT USING "tO

PRINT USING "#,K"!

ihi. " LOW VALUES 1

(hours)

(cumec)"

(cumec) <h6urs>"

6630 ELSE 20-6640 PRINT6650 END IF6660 PRINT USING " <m>6670 IF Rh_bc«»2 THEN6680 PRINT USING <a>6690 ELSE 6?e0 PRINT USING -//"6?$0 END IF6?S0 FOR 1=1 TO II6 7 3 0 PRINT USING '•#, 3 X , 3D, 2 <6«, 3 U . 3D , 2X , 3 D . 3D > "; I, Ymax< 1, 2 ) , Ymai<< 1,1 >, Qmax<1,2), Gma>i< 1,1)6740 IF Rh bc=2 THEN6750 PRINT USING "7X,3D,35,2X,3D.3D";Ymin(I),QmIn<I>6760 ELSE6770 PRINT6780 END IF6790 NEXT I 6860 PRINTER IS 16 6810 RETURN6820 I **************************************,»********************************6630 I PRINT MASS BALANCE RESULTS Level 36640 I ***********************************************************************6650 Print balance: I6860 PRINTER IS 06870 PRINT USING _ _ _ _ _ _ _ _

(1000m3) '.Vol

6900 PRINT USING "3X,21 A,7DZ. 3D";" Inflow (1000m3> :",Q9um_ln/1000 6918 PRINT USING "#,31A,7D2.3D";" Final Volume (I000m3) :",Vol_end/10006920 PRINT USING U3X, 21 A, ?DZ. 33, " Outfley (1000*3) ; Qsum_out/10006930 Vol_ehange=Vo1 end-Vol start6940 PRINT USING "OiX, 120,24%, 12A,/";"________________ "6950 PRINT USING 31A,7DZ.3D";11 Increase 1n Volume <1000m3> i",Vot_change/l

6960 PRINT USING "3X,2im,7DZ.3D,/";

6970 PRINT USING " 31 A,7DZ.3DX,ft"5"-Qsum out))/Vol change*100,6960 PRINT USING " / / , K , _______

" Net Inflow <1000m3) :11, <Qsum_1n-Qsu»_< Error :", (Vol_charige-(0&u;

PRINTER IS 16 7000 RETURN7016 I **********************************************************************7020 ! HYDROGRAPH DATA INPUT Level 37030 I **********************************************************************

720072107220

Hydro„lnput:DIM A<100,2>PRINT PAGEPRINT " HYDROGRAPH INPUT I PRINT "POINT

I INPUT OF A H1'GRAPH

LJtlPUT "ENTER 7INE(HRS> i, INFLOW(M3/S)- EXIT IF E d H # = ,,l‘

COSUB S t r i n g s A < I , l ) - V A L < S t r # < $ ) )A <I ,2 >»V AL (S t r# <8> >PRINT USING 7 1 9 0 ; i , m < I , l > , F K I , 2 >IMAGE 2>i ,DD,16X,DD.DD,16X,5D.DDD

END LOOP S e t 3 = I - l G08UB E d i t IF E d t t $ C l , 1 1 - RETURN

<C0NT WHEN F I N I S H E D ) " ,E d i t *

Y" THEN GOSUB Hydro_edlt

INPUT OF MUL'-PUE INFLOW H'1 GRAPHS

7270 j »»#*>**!.*»#*#*»*»»**#*#»»**#»#*»#***«*#»*««*****»*«#««***»**«*»***•*****

7290 Hydro_Adfl:7306 PRINT PAGE7310 PRINT "NOTES TO ADDING OF H•'GRAPHS: "7328 PRINT-"---------------------- "7330 PRINT LIN(I);"1. ALL H'GRAPHS TO BE ADDED MUST HAVE DATA GIVEN WITH THE S ARE TIME INCREMENTS."7340 PRINT "2. ALL LAG TIMES TO BE A WHOLE NO. OF TINE INCREMENTS."V358 DIM Hydrc.,l00,2>,Lag<t0>,T1ffie<I0>7360 MAT Hydro=ZER7370 Set3_m&x“07380 INPUT "HOW MANY H'GRAPHS DO YOU WISH TO ADD ?",Hhydro7390 FOR 2=1 TO NHydro7400 DISP "IS H'GRAPH NCI. ";Z;"ON A FILE?";7410 INPUT J4$7420 IF J4$="Y" THEN 75207430 IF J4#=I,N" THEN 74707440 BEEP7450 INPUT "ENTER Y IF H'GRAPH ON FILE OR N IF NOT I! ",J4*7460 GOTO 74207470 GOSUB Hydro Input7480 GOSUB Hydro_stere7490 DISP "ENTER LAG TIME FOR H'GRAPH NO. ";Z;7500 INPUT Lag(Z)7510 GOTO 75607520 • DISP "ENTER FILE NAME FOR H'GRAPH NO. ";Z;" AND LAG TIME";7530 INPUT •nr1o*,Lag<Z>7540 ASSIGN #3 TO *7550 GOSUB Sur#_fi led7569 IF Lag<Z)=0 THEN JJ=07570 IF Lag<zn0 THEN Jj = INT<Lag<Z)/T>7580 TI«ie<Z)=<A<Set3,1>-AC1,l>>7<Set3-l>7590 IF Z<2 THEN 76107600 IF Time(Z>OTW*<Z-l> THEN 96407610 T=Tise<Z)7620 FOR J=I TO Set37630 A<J,l)=A<J,l)+Lag<Z>7640 IF Set3 max<Set3 THEN HydroCJ,I>=</-1)#T7650 Hydro< J+Jj,2)=Hydro<J+JJ,2)*A<J,2>7660 NEXT J7670 IF Sets max<SetS+Jj THEN Set3 #ax=Set3+Jj7680 NEXT Z7690 Set3=Set3_max7766 MAT A*Hydro7710 PRINT PAGE7728 GPSUB Hydro print7730 RETURN

7756 I ********************************************4***************************

I EDITS INFLOW H'GRAPH

LINPUT "ENTER NO. OF POINT TO BE CHANGED - t CONT WHEN FINISHED >",I

EXIT IF 1*=""I=VAL<I$>EdH»=" ”8,VAL$<A<I, !))&" ,DISP "TIME ,INFLOW for Po1n

EDIT Edit*GOSUB Strings A<1,1 >=VAL<Str*<1> >A < I ,2 ) = V A L ( S t r * ( 2 > )GOSUB Hydro print

END LOOP RETURNI ************************************************************************

"& V A L $(A (I,2 )>" i I i “ d i s p l a y e d b e lo w , Cha nge i

7 940 H y d r o _ s to r e : ! SUBROUTINE TO STORE INFLOW H'GRfiPH ON DISC7 950 INPUT ENTER NAME OF FIL E IN WHICH YOU WISH TO STORE H•'GRAPH D A T A ",In f l

7960 RED1M A < S e i3 ,2 )7 970 R e c o rd s® IH T < 8 * 2 * S e t3 /2 5 6 )+ I 7 980 CREATE I r t f l o $ , R e c o rd s7990 ASSIGN #6 TO I n f l o *3 000 PRINT #6;A <*>0 010 ASSIGN 66 TO *902 0 RETURN

804 0 ! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

806 0 H y d r o _ p r m t : I PRINTS INFLOW HYDROGRAPH807 0 PRINTER IS 16808 0 PRINT PAGE8090 PRINT " HYDROGRAPH : " ,L IN ( 2 )8100 PRINT "POINT TIME FLOW"8110 PRINT " Ho. <h> (c u w e c >" . L IH (1>8 32 0 FOR 1=1 TO S # t38 13 0 PRINT USING 8 1 5 0 ; I , A ( I , I ) , A ( I , 2 )8 14 0 NEXT I8 150 IMAGE 2X ,D D ,16X ,D D .BD ,16X ,5D .D D D8 16 0 RETURN

8 18 0 ! **********************************************************************8.260 H ydro r e a d : ! READS H''GRAPH FROM DISC8 21 0 INPUT "NAME OF H'GRAPH F IL E ? " , I n f 1o$8 2 2 0 S u m _ fM e d : I SUBROUTINE TO READ INFLOW H'GRAPH FROM DISC823 0 REDIM A (1 0 0 ,2 )824 0 ON ERROR GOTO 3 210825 0 ASSIGN #3 TO I n f i o S826 0 OFF ERROR8270 S e t3 = l828 0 LOOP8290. READ # 3 ; A ( S e t 3 , l ) , A ( S e t 3 , 2 )830 0 ON END 43 GOTO 82308310 S e t3 = S e t3 + l6 320 END LOOP6 330 ASSIGN #3 TO *8348 S e t3 = S e i 3 - l8350 REDIM A < S e t3 ,2 )8360 INPUT "DO YOU WISH TO MULTIPLY ALL INFLOW ORDINATES BY A CONSTANT ? (Y /N )" ,E d H #8370 LOOP8360 EXIT IF < E d U * t l , OR CEdl t ȣ 1 , n = "N ")8390 BEEP8480 INPUT "ENTER Y IF YOU WISH TO FACTOR THE INFLOW H'GRAPH OR N IF NOT I"

8410 END LOOP842 0 IF E dH S C 1 , 13®“ Y“ THEN843 0 INPUT "ENTER THE CONSTANT", C o n s t844 0 FOR 1=1 TO S e t3 >8 4 5 0 R < I ,2 ) = A < I,2 > » C o n s t846 0 NEXT I8470 END IF848 0 RETURN

850 0 I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 8 5 :0 D i s c r e t e _ h y d r o : ! DISC.TETIZES HYDROGRAPH8520 MAT Q _ d isc re te = Z E R8530 REDIM 0 d i s c r e t e < 5 0 0 0 )8540 T s te p = IN T ( A (S e t3 ,1 > /D t1 )+ 1855 0 N n= IN T < T m ax /D tl> * l856 0 FOR 1=1 TO S e t 3 - t857 0 S t e p l = I N T ( ( A ( I + l , 1 ) - A ( I , 1 ) ) / D t l )858 0 Q s te p = A < I , I ) / D t 1 + 1859 0 FOR K = Q step TO S t e p l* Q s t e p860 0 Dq = (A ( I + l , 2 ) - A ( I , 2 ) ) / ( A ( I + l , l ) - A ( I , D )861 0 Q_d i s c r e t e <K) = < K -O step )* D q * D t1 + A < 1 ,2 )

NEXT K - 1 2 3 -

S e tr3 _ o l d = S e t3S e t3 = T s t-e pG d i s c r e t e < S e t3 ) » A < S e t 3 o l d , 2 )REDIM Q _ d is e re te < N n )

8700t * » * * » » * # * * * # * * » * » * * * * # * # * * * * * * # » * # ♦ * « # * # * * * * # * * * » * # # # # * » * * * # * * * * * * * * * « * »

6720I **********************************************************************

S t r i n g s : t BREAKS BOHN ft COMMA-SEPARATED STRING! INTO COMPONENT STRINGS

C o m m a = P O S (E d it$ ," ,">IF Corei»a<>0 THEN S l r$ < I )= E d i %$C 1, Comma]IF Comi»a=0 THEN S t r$ < 1 )= E d i t *I j= 2

EXIT IF Commae 6E d it t -E d itS C C o m m a + IlC om B ia=PO S <E dit$,u ," >IF ComroaOO THEN

S t r # < I j ) = E d i t* C I ,C o 6 im a - l ]

S t r $ < I j ) = E d i t *

I j = U * IEND LOOP

E d i t : I CHECKS IF ANY EDITTING TO DISPLAYED1 DATA IS REQUIRED

INPUT "DO YOU WISH TO MAKE ANY CHANGES TO THE ABOVE DATA ? ( Y / N ) " ,E d i t

EXIT IF < E d i t* C i , i 3 = “Y ") OR < E dH 6E 1 , n = " N ")

D ISP "ENTER YES OR NO I I I "WAIT 2 800

END LOOP

| ***********$**$*$*$$$**$*$*$*******$*$***$$*$$**$***$$******$*$$*$******1 STORING AND READING IN IT IA L CONDITIONS FROM DISC

I n 1 t s t o r e : 1INPUT “ ENTER NAME OF FIL E FOR STORING IN IT IA L DATA", I n i t *R e c o r d s = N $ e c t1o n s * 2 * S /2 S 6 + lCREATE I n i t S ,R e c o r d sREDIM O s t a r t < N s e c t 1 o n s ) ,Y s t a r t ( N s e c t I o n s )ASSIGN #5 TO Ini **PRINT #5JQ s t a r t<*), Y s t a r t<*)ASSIGN #5 TO *RETURN1 •****##***#****#**«****#******#»*»»**»«*******»**«•*******#*****»#***»»

) ***********************************************************************I n l t r e a d : 1

REDIM Q s t a r t ( M s e c t i o n s ) , Y s t a r t ( M s e c t I o n s )INPUT "ENTER NAME OF FIL E IN WHICH IN IT IA L DATA IS STORED", I n i t $ASSIGN r'S TO I n i t *READ # 5 j Q s t a r t < *> , Y s t a r t < *>ASSIGN #5 TO *

I ************************************************************************I MASS STORAGE DEVICE L e v e l 31 ************************************************************************

S t o r a g e d e v i c e : IE d i t$ = P a th $ & D e v ic e #IF E d i t f O " " THEN

R em ote < R )" ,D e v i

.h e DIRECTORY PATH " ,P a th *

9 5 7 0 P 'o t_ h y d r o :S U B PI o t_ h y d rc i< N n , Dt 1 , Q_1 h s< * > , Q_rhS'

LOCATE 1 3 ,9 5 , Q m ax-0

MAT SEARCH Q lh5,M IN JQ m 1n MAT SEARCH Q rhs,M A X ;Q m axl MAT SEARCH 0 rh s ,M IN ;Q m in l

IF Tmax<=4 THEN X s te p = ,5 IF Tmax>4 THEN X s te p = l

Yrange=Ymax-Ynil nIF Yrange/Ystep<4 THEN Ystep=Ystep/2 IF Xsca1e/Xstep<4 THEN Xstep=Xstep/2 AXES Xstep, Ystep,0,Y«iln,4,5,2 MOVE 6,Q_lhs(I)FOR 1=2 TO Nn

LINE TYPE 5,1 DRAW <I-$ >*Bt1,Q Ihs(I)

161801019610200

i!§

NEXT I -1 2 5 -MOVE 0 ,0 r h s C D LINE TYPE 1 FOR 1=2 TO Nh

DRAW r h s < I )

I Y-AXIS LABELS CSI2E 3 . 5 , . 6

FOR Y-Ywln TO Ymex STEP Y s te p MOVE 0 ,YLABEL USING "K ,X ";Y

I X-AXIS LABELS

FOR X=0 TO X s c a H STEP X s te p MOVE X .Y re tn - Y r a n g e /ie e

1 Y-AXIS TITLE

LBIR 90

MOVE 2 ,5 8LABEL USING " # ,K " ; Y H t l e $

' X-AXIS TITLE

MOVE 6 0 ,5 .LflBEL USING -# ,K " i" T IM E (HOURS)"

I LEGENDMOVE 4 8 ,9 5

! LABEL “Run " ;R u n MOVE 5 0 ,9 7 LINE TYPE 5 ,1 DRAW 6 0 ,9 ?LINE TYPE 1

CSI2E 3 . - 5 , .6LABEL " UPSTREAM " fc Y tI t 1e * C 1 ;5 3 MOVE 5 0 ,9 2 DRAW 6 0 ,9 2

LABEL " DOWNSTREAM '•&Yt U 1 e » C l ;5 3 DUMP GRAPHICS EXIT GRAPHICS PRINTER IS 16 STANDARD

SUBENDI ! M I I I I M I I M I I 1 III I M I ! II! M I ! I I! 1 I ! 1 I ! ! ! .....11 I! ! I ! I..... Ml 11 H I IIS e c U o n _ ta b te Z S U 8 S e c t Io n _ t a b ? I , N x ,W si, B , A re a ,K su m ,P su m , A1 p h a ,B e ta >

OPTION BASE 1COM Di s t a n c e <■»> , Np< *> , L e v e l 1 < * ) , Ma«nC*>DIM A <20>eK < 2 0 > ,P < 2 0 ) ,R < 2 0 > ,X s < 2 0 ) ,Y s < 2 0 ) ,S tA rtx < 1 0 ) ,E n d x < 1 0 > ,X x < 2 0 > IF FRACT<Ws1)< .0 0 0 1 THEN Ws1»INT<W*f>A rea=K sum =Psum =0 A 8=K8=Ps=W d=Beta=B=0 A1phA _ su a= B eta_ su m = e S t a r t= E n d = 0 MAT A=<0>MAT K=<0)MAT S ta r tx = 2 E R MAT Endx-ZEh.FOR J= 1 TO N p < N x )-l

X 1 = D 1 s ta n e e < I , J )

I

MANNING N GIVEN AT A POINT

D y=H st-Y 2C x= D y*< X l-X 2> /< Y l-Y 2)

D y= H si-Y lD x=By*<X 2-X l>/'<Y 2-Y iX@(J+I>=X!+BxY »< J+ I)= Y l+ I)y

L y "Y s< J+ I> -Y s<P<J>=SQR<Lx*Lx+Ly*Ly>

B eta_sun i= B e ta_auB i+ K ( J ) A2 /fl< filpha_sugi=A !pha_8U Fi*K <J>A3 /f t< J ) '- 2

A1phe=AI pha_sumy, <Ksu«nA3 /R r e a A2 )

B e ta = R lp h a = l

APPENDIX B2 : PROGRAM LISTING - BACmATA11 HYBRO/BACKDATFi: REMOTE"

UPDATE 1 9 8 5 -0 8 -0 1 FJGG OPTION BASE 1 COM P a th * E 5 Q 3 , D evi e e$ C 5 0 ]COM R e a c h < 6 0 ,2 0 , 3 > , N p<69>, L ( 6 0 ) , D a ta DIM C |» < 6 0 > ,B p < 6 0 > ,B e a c h < 6 0 ,2 0 |3 )DIM E d l t* £ 8 0 ]E d i t v a lu * = 0

2 0 0 PRINTER IS 163 2 8 PRINT PAGE;" BACKWATER PROGRAM - CROSS SECTION DATA INPUT" . LIN<2>2-10 PRINT CH Rt<27>&"1"266 GOSUB y to r a g e _ d e v i c e280 LOOP300 LINPUT " E n te r t h e REACH Number -< CONT I f FINISHED > " ,N r$320 EXIT IF N r» = ” "340 Nr=VAL<Nr*>360 GOSUB R e t r i e v e380 GOSUB O p t io n s

END LOOP

I *********************************

500 O p t io n s ; I AVAILABLE OPTIONS IN THIS PROG520 LOOP540 PRINT PAGE," DATA FOR REACH .........560 PRINT " 1 . ENTER CROSS SECTION DATA " , L1NC1 >5 0 0 PRINT “ 2 . PRINT HARD COPY OF CROSS SECTION DATA", L IN <1>600 PRINT “ 3 . EBIT CROSS SECTION DATA", L IN <$>620 • PRINT " 4 . RUN BACKWATER PROGRAM WITH DATA FOR REACH ,!;N r ,L IN < l>648 PRINT " 5 . ENTER A NEW REACH "6 6 6 INPUT "ENTER OPTION < 1 , 2 , 3 , 4 o r 5 ) " , Op660 SELECT Op700 CASE 1720 GOSUB D a ta746 GOSUB S t o r e760 CASE 2780 GOSUB P r i n t o u t l800 CASE 3820 GOSUB D a ta e d i t840 GOSUB S t o r e860 CASE 4661 Oatasi8 8 0 LOAD "HYDRO/BACKWATERlREMOTE",1900 CASE 5920 EXIT If/ Op-5940 CASK ELSE960 I6EEP980 UNPUT " ENTER AN INTEGER BETWEEN 1 AND 5 TO INDICATE OPTION 1 T " ,0 p1000 GOTO 6801020 END SELECT1040 END LOOP1060 RETURN

1100 I **********************************************************************

1140 D a ta : I INPUT OF CROSS-SECTION DATA IN A REACH1160 PRINT PAGE1180 PRINT “ POINT DISTANCE LEVEL MANNING "1200 PRINT * No. <«> (m> N "1220 PRINT CHR$<87)8."1"1240 N p_*ax»0

1280 LOOP1308 D a t a ! : I START OF BRANCH FOR RE-ENTERING A CROSS SECT

1320 PRINT U N C O , " Cf-oss Sec11 on No, 11; I; "“.LIN< 1 >1340 IF I>i THEN1360 DISP "CHANNEL LENGTH BETWEEN SECTIONS";1-1J"AND";I $" CCONT IF I-1i"IS LAST SECTION1380 LINPUT " ".Edit*1480 EXIT IF Edit$=""1420 L<I)=VAL<Edit*)1440 END IF

1480 LOOP1500 LINPUT "ENTER DISTANCE,LEVEL .MANNING N - < CONT WHEN FINISHED WITH THIS SECTION >\Edit*1520 EXIT IF Edlt*="“1340 GOSUB Strings1560 Reaeh<I,J,l)=VAL<Str*<l))1500 Reachd, J,2)=VAL(Str*(2))ISUQ Reach<I,J,3>=VAL<Str$<3>)1620 IMAGE 3X,DD,11X,4D.D, 8X,3D.DD, 9XD.DDD1640 PRINT USING 1620;J,Reach<I,J,1>,Reach<I,J, 2>,ReachCI, J,3>1660 J=J*t1680 END LOOP1700 Np<I>=J-l1720 IF Np<I)>Np max THEN Np_max=Np(I)1740 IF I>1 THEN1760 PRINT "DISTANCE BETWEEN SECTIONS “JI-IJ" AND*;I;11 IS";L(I)1780 END IF1600 PRINT LIN<2>1820 GOSUB Edit1840 IF Edit*Cl,13="Y" THENI860 GOSUB Data editl1880 END IF1900 IF Edit_value=l THEN 20081920 Cum_np=Cum_np+Np<I)

1960 END LOOP 1980 Nx=I-l 2800 RETURN2048 ! •*«»«#*###****####»**»

2000 Data editl: I EDITTING OF DATA2100 IF 7>1 THEN2120 Edi t*=VRL*<L<I> >2140 DISP * CHANGE THE DISTANCE BETWEEN SECTION";!;" AND SECTION"JI-I}2160 EDIT Edit*2180 L<I>=VAL<Edit*)2200 END IF 2220 LOOP2240 DISP "WHICH POINT DO YOU WISH TO CHANGE IN CROSS-SECTION";1$" (CONT WHEN FINISHED2260 LINPUT Edit*2280 EXIT IF Edit*-""2300 Cp=VAL<Edit$>2328 Edit*-" “8,VAL*(Reaeh<l,Cp,ii)&Com*l,VAL*<Reaeh<I,Cp,2))8.Com*&VAL*(Reach<I,Cp,3>>2940 DISP “ CHANGE DISTANCE,LEVEL OR MANNING N AT POINT ";Cp;2360 EDIT “ ".Edit*2380 GOSUB Strings2408 Reachd.Cp, l)=VAI.<Str*<l>)2428 Reachd.Cp, 2 >=VAL<Str*<2>)2448 Reachd.Cp, 3>«VAL<Str*<3>)2468 GOSUB Printout2480 END LOOP 2500 RETURN

2340 I

2580 Store: I STORES DATA ON DISC.2680 REDIM Reach<Nx,20, 3>, Np(Nx>,L(Nx>2620 Records-1NT< <8*60*Nx+16*Nxlx256>^l

2640 ON ERROR GOTO 2780 2660 Na*ie*="Rea<"8,VAL$<Nr>2680 CREATE Naae$,Records 2700 ASSIGN #1 TO Nai#e$2720 PRINT ttliCum np,Nx,Np<#>,L<*2740 ASSIGN * TO #t

GOTO 2920

PRINT ERRM#IF ERRN=54 THEN

DISP “ENTER A NEW REACH NO. OR PURGE FILE “{Name*; INPUT Hr GOTO 2660

END IF OFF ERROR RETURNI **********************************************************************

3020 Pr1nteut!PR INTER IS 16 I PRINTS OUT CROSS SECTION DATA 3040 PRINT TAB<5>;"REACH NUMBER";Nr 3060 PRINT TAB<5>{ " = ===== = «■ = = ==== = = “3080 PRINT TAB<5)|“CROSS";TAB<20>{“POINT";TAB(35>{"DISTANCE";TAB<50>;"LEVEL"{TA B<65>{"MANNING"3100 PRINT TAB<4);"SECTION";TAB(20>{"NUMBER";TAB(69);"N"3120 PRINT TAB<4> j “ = ===== = "{ TAB<20> { "== = = = =»"; TAB<35> { "==«— =='1 {TABC50) { " = " 5TAB(6S){ "saaost-a"3140 PRINT LIN(2)

PRINT USING 3180;!,1,Reach!I,1,1),Reach!I,1,2),Reach!I,1,3)IMAGE 5Xf3Dil3K,DD,10X,5D.D, 8X.3D.DD, 9XD.DDDFOR J=2 TO Np(I)

PRINT USING 3240JJ,Reach!I,J,1),Reach!I,J,2>,Reach!I,J,3>IMAGE 21X,DD,10X,5D.D, 8X,3D.DD, 9XD.DDD

IF 1=1 THEN 3320PRINT "DISTANCE BETWEEN SECTIONS "{1-1;" AND"{I{" IS"{L!I)PRINT LIN12)

324032603268

! READS DATA FROM DISC

I ************************************************************************

3440 Retrieve:3460 Name reach$="Reac"tVAL*!Nr>

ON ERROR GOTO 3600 ASSIGN #3 TO Name_reach*READ #3;Cum np,Nx REDIM Reach!Nx,20,3>,Np<Nx),L!Nx>READ tt3;Np!*>,L!*),Reach!*)ASSIGN #3 TO *OFF ERROR RETURN

3700 Dat 3720 Edit

| *********************************************************************

\LIN!2>

t: I DATA EDIT OPTIONS

L i . -PRINT PAGE," EDITTINC OF DATA FOR REACH ":Nr.PRINT " 1. INSERT A NEW CROSS SECTION ",LIN!1)PRINT “ 2. CHANGE EXISTING DATA",LIN!2)PRINT " 3. DELETE A CROSS SECTION ",LIN!2>LINPUT "ENTER OPTION ! lor 2) - tCONT WHEN FINISHED >",Edlt$

EXIT IF Edit*=Op edit=VAL<Edit<) SELECT Op_edit

INPUT "ENTER No.

GOSUB Insert

OF CROSS-SECTION BELOW NEW SECTION ",Jk

•UCH CROSS-SECTION DO YOU WISH TO CHANGE ? ( CONT WHEN

• U WANT TO CHANGE THE WHOLE SECTION ? <Y/N)",Edtt*

GOSUB Datal

4080 LINRUT ’FINISHED >",Ed1t*

EXIT IF EdH.r 4120 I=VAL(Ed1t414A INPUT "DO4160 LOOP4180 EXIT IF < E d i mi,l]»HY"> OR (Edi t$E 1, 1 ]»"N")4200 DEEP4228 INPUT "ENTER Y IF YOU WANT TO CHANGE THE WHOLE SECTION ORN IF NOT ",Editt END LOOP

IF Edn*="Y" THEN GOSUB Datal

GOSUB Data edit!END IF

END LOOP

LINPUT "WHICH CROSS-SECTION DO YOU WISH TO DELETE ? ( CONT WHENFINISHED >",Edit*

45004520

468047004720

EXIT IF Edit#"""I i »Vf iL<Ed i t*>L < I i* l ) = L < :H > * L < I i> i>FOR I .=Ii TO Nx-l

FOR J=1 TO Np<I*l)Rcachd, J, l)=Reaeh<I*l,J, 1) Reachd, J,2)=Reach(Hl,J,2> Reachd, J,3>=Rea6h<I + l8 J.S>

NEXT JNp<I>=Np<1+1>L< I >"L<I + 1)

Nx=Nx-l REDIM Rea

END LOOP CASE ELSE

END SELECT END LOOP RETURN !

h(Nx,20,3),Np(Nx),L(Nx)

i ***********************************************************************

4900 PMhtout 1!PRINTER IS 0 IPRINTS OUT CROSS SECTION DATA 4920 PRINT TAB(10);"REACH N U M B E R N r 4940 PRINT TAB<10>;"=======b=b h b===b "4960 PRINT TAB(10);"CROSS";TAB(20);" REACH"}TAB<32>$"POINT”;TAB<42);"SECTION"$ TAB<55>;-GROUND";TAB<65>|"MANNING"4980 PRINT TAB<9>5"SECTION"5TAB<20>;"DISTANCE";TAB<32>;"NUMBER";TAB<42>;"DISTAN CE";TAB(55>;"LEVEL";TAB(69);"N"5000 PRINT TAB<9>;"======="JTAB<20>;"====■===";TAB(32>;"======";TAB<42);"==■====="i TABI55);"======";TAB<65);"======="5020 PRINT LIN(l)5021 Disffl5040 FOR 1 = 1 TO N'(5041 DlSt=DIst +L<i)5060 PRINT USING 5080;I,Dist,1,Reach<I,1,1>,Rea5h<I,1,2),Reach<I,1,3)5080 IMAGE 10X,3D,7X,4D.2B,7X,DD,6X,4D.D, 6X.3D.DD, 4XD.DDD5100 FOR J=2 TO Np<I)5120 PRINT USING 3140jJ,ReachCI,J,1>,ReachCI,J,2>,Reach<I,J,3)5140 IMAGE 34X,DD, 6X,4D.D, 6X,3D.DD, 4XD.DDD5160 NEXT J5220 PRINT LIN<1>3249 NEXT I 5260 PRINTER IS 16 5280 RETURN

5326 I **********'****t*******************************************************

5360 1nsert:MRT Cp=Hp 5360 MAT Bp-L 5408 MAT Beach=Rea.ii5 4 2 0 R E D I M R e a c h < N 1 , 20, 3 > , N p < N x + i > , L < N x + l ) , C p ( N x + i ) , B p ( N x + 1 ) , B e a c h ( N x * I , 2 0 , 3 ) 5 4 4 0 F O R I - J j + 1 T O N x + 15460 FOR J=1 TO N p ( I - l )5480 B e a c h < I , J , l > = R e a c h < I - I , J , l )5506 B e a c h <I , J , 2 > = R e a c h < I - l , 3 , 2 )5520 B e a c h < I , J , 3 > = R e a c h < I - $ , J , 3)5540 NEXT J5560 Cp <I >” Np( I - 1)5580 Bp( I > = L < I - ! ?5600 NEXT I5620 MAT Reach-Beach5640 MAT Np=Cp5660 MAT L=Bp5680 Nx=Nx+t5F00 RETURN5720 STOP5740 I **********************************************************************

5780 Strings! I BREAKS BONN A COMMA-SEPARATED STRING3600 I INTO COMPONENT STRINGS5820 Coi»sa=POS<Edi t«, >5840 IF CommaOe THEN Str*< I )=EdH *E i, Comma]5860 IF Cemhia=0 THEN Str*< 1 )=Edit*5880 11=25908 LOOP5920 EXIT IF Comma=05940 Edi t$=Edi tSECemma*135960 Comma=POS<Edlt#,",")5980 IF Cemma<>0 THEN6000 Str$(Ii>=EditSCt,Comma-I]6028 ELSE6040 Str*<I1)=Edlt#6060 END IF6080 11=11+16100 END LOOP 6120 RETURN6140 Edit: I CHECKS IF ANY EDITTING TO DISPLAYED6160 I DATA IS REQUIRED6190 Edi t*=" "6200 LOOP6220 INPUT "DO YOU WISH TO MAKE ANY CHANGES TO THE ABOVE DATA ? <Y/N>11,EdltS6240 EXIT IF <Edit»tl,13=’"Y‘l) OR (Edlt$Cl,l] = "N")6260 BEEP6280 DISP "ENTER YES OR NO III"6360 WAIT 20006320 END LOOP 6340 RETURN6360 I ************************************************************************ 6380 I MASS STORAGE DEVICE Level 36408 I ************************************************************************

6420 Storage_devlee: I6440 Edit#=Path*&DevIce#6460 IF Edi t$< >"" THEN6480 EDIT "CURRENT MASS STORAGE IS (CONT If OKAY)",Edit* 6500 IF Ed1t#="“ THEN 66806520 Co:on=POS(Edlt$,":">6540 Dev1ce*=EdltSCColonl6560 IF Colon>l THEN Path*=EditFI1,Co Ien-U6500 ELSE 6600 LOOP6620 INPUT "Local <L) or Remote <R)",L vice*6640 EXIT IF <Device$=“L") OR (Devlce*="R">6660 BEEP6660 DISP " L OR R EXPECTED FOR ENTRY :

WAIT 1000 EH3 LOOPIF Dev1ce#»"R" THEN

Devi cete"!REMOTE" -INPUT "Enter the DIRECTORY PATH ",PathsLINPUT "Enter the Local device Address < eg, :H7,0,1 )",Device$ Path*-"*

MASS STORAGE IS Path#8-Device*RETURNI t***********************************************************************

APPENDIX C

SWARTVLEI ESTUARY

CROSS SECTION DATAGROUND

1 5 = 9a6.0i 4 . 0

55.099.0 .

1 1 1 . 0

3 8 2 0 . 0 6

3940.00

4350.08

4630.06

4830.60

5190.06

5350.00

^ # 9

APPENDIX C2 : SWAKTVLBI ESTUARY - RESULTS OF TIDAL RUN

FINITE DIFFERENCE FLOW BNALYS1S using the COMPLETE DYNAMIC EQUATIONS

DATA FILES :Cross sections : R<?ac2LHS Boundary Conditions: Inflow hydrographs QTWO RHS Boundary Condit iohs: Tidal variation:

Mean sea level ! 3,29 mM2 efflp!i t ude ! ,680 mS2 amp) ttude i .35 tnPhase Difference! 8 s Time of Tide $340.6? h

Initial Conditions : INIT2_363

THETA ! .6 PSI ! .5TIM'S STEP : 0. 10 hours

SECTION 3 SECTION 5 SECTION 19 SECTION 36 SECTION 44s i a s i SJAGE FLOW st ag e FLOW FLOW

- 3 , 8 7

- 3 S .9 4-41.12- 3 0 .9 3

3 .9 12 2 .0 92 0 .7 91 6 .1 7If.2114.70113.36

. ?3 - 6 .9 2-218 .32

-3 5 .2 (5 - 4 3 .3 2- 3 4 .5 6

MAXIMUM STAGE MNP FLOW VALUES

k -

20^80020,600

SB.300

14.000

3.591

3.5el53.579

UPSTREAM FLOW DOWNSTREAM FLOW

5 -10

2 - 2 0

25 30TIM E (H O U RS)

. #

I

- UPSTREAM LEVEL

— DOWNSTREAM LEVEL

25 3815 2 0TIM E (H O U R S)

Inflow <1806m3>

64.9554,5454.50

54.4054.9456,30

J

w #

REFERENCES

1. Abbott, M. B. and Perwey, A. Four Point Method of Characteristics, Journal of the Hydraulics Division, ASCE, Vol 96, No. HY12, pp. 2549 - 2564.

2. Abbott, M.B. (1975). Method of characteristics, In :Mahmood, K and Yevjevich, V. ed. Unsteady Flow in Open Channels, Fort Collins, Colorado : Water ResourcesPublications, Vol. 1, pp. 63 - 88.

3. Abbott, M.B. (1979) Computation Hydraulics, Elements of the Theory of Free Surface Flows, Pitman Publishing Limited, London.

4. Akan, A.O. and Yen, B.C. (1981). Diffusion wave flood routing in channel networks, Journal of the Hydraulics Division, ASCE, Vol. 107, No. HY6, pp. 719 - 732.

5. Amain, M and Fang, C.S. (1970) Implicit Flood Routing in Natural Channels, Journal of the Hydraulics Divi­sion, ASCE, Vol 96, No. HY12, pp. 2481 - 2500.

6. Chaudhry, M.H. (1979). Applied Hydraulic Transients, New York : Van Nostrand Reinhold Company,

7. Cunge, J.A. (1969). On the subject of a flood propa­gation computation method (Muskingum method), Journal of Hydraulic Research, International Association for Hydraulic Research, Vol. 7. No. 2, pp. 205 - 230.

8. Cunge, J.A., Holly, P.M., and Verwey, A. (1980). Practi­cal Aspects of Computational River Hydraulics, Pitman Publishing Limited.

9. Read, D.L. (1978). National Weather Service Opera­tional Dynamic Wave Model, National Weather Service, U.S. Department of Commerce.

10. Hayatni, S. (1951). On the Propagation of Flood Waves,Bulletin No.l, Disaster Prevention Research Institute, Hyola University, Japan.

11. Heggen, R.I. (1983) Flood Routing on a Small Computer,Civil Engineering, ASCE, Vol.53, No.3, pp 63 - 66.

12. Huang, J. and Song, C.C.S. (1985). Stability of dynamic flood routing schemes, Journal of Hydraulic Engineering, ASCE, Vol. Ill, No. 12, pp. 1497 -1505.

13. Huang, Y.H. U978) Channel Routing by Finite Diffe­rence Method, Journal of the Hydraulics Division, ASCE, Vol.104, No. HY10, pp 1379 - 1394.

14. Huizinga, P. (1984). Application of the NRIO 1-DHydrodynamic Model to the Swartkops Estuary, CSIR Research Report 560, Stellenbosch.

15. Joliffe. I.B. (1984). Computation of dynamic waves inchannel networks, Journal of Hydralic Engineering, Vol. $10, No. 10, pp. 1358 - 1370.

16. Koussis, A. (1976). An approximate dynamic flood routing method, Proceedings of the InternationalSymposium on Unsteady Flow in Open Channels, Paper LI, Newcastle-upon-Tyne, England.

17. Li, R.M., Simons, D.A., and Stevens, M.A. (1975).Nonlinear kinematic wave approximation for water routing, Water Resources Research, Vol. II, No. 2, pp. 245 - 252.

Liggett, J.A. and Gunge, J.A. (1^75). Numerical methods of solution of the unsteady flow equal:ions, In: Mahmood, K and Yevjevich, V. ed. Unsteady Flow in Open Channels, Fort Collins, Colorado : Water Resources Publications, Vol. 1, pp. 89 - 182.

McMahon, G.F., Fitzgerald, R and McCarthy, B. (1984), BRASS model: practical aspects, Journal, of WaterResources Planning and Management, Vol. 110, Ho. 1, pp. 75 - 69.

Miller, W.A., and Cunge, J.A., (1975). Simplifiedequations of unsteady flow. In; Mahmood, K and Yevjevich, V. ed. Unsteady Flow in Open Channels, Fort Collins, Colorado; Water Resources Publications, Vol.l, pp. 183 - 257.

National Research Institute for Oceanology (1975), Hydrographic Survey of Sedgefield Lagoon, OS IS Report C/SEA 75/13, Stellenbosch.

National Research Institute for Oceanology (1978), Hydrographic Study of the Swartvlei Estuary, CSIR Report 0/SEA 7805/1, Stellenbosch, (in Afrikaans).

Peterson, W.C. and Verhoff, F.H. (1982) Muskingum­like Approximates for Water Routing, Journal of the Hydraulics Division, ASCE, Vol.108, No. HY11, pp. 1387 - 1397.

Ponce, V.M., Li, R.M. and Simons, D.B. (1978). Applica­bility of kinematic and diffusion models, Journal of the Hydraulics Division, ASCE, Vol. 104, Vo. HY3, pp. 353 - 360.

Price, U.K. (1973). Flood routing methods civexa, Proceedings of the Institution Engineers, Part 2, Vol. 55, pp. 913 - 930.

Pri<; R.K. (1974) Comparison of Four Methods for Flood Routing, Journal of the Division, ASCB, Vol.. 100 No. HY7, pp. 879 -

for British of Civil

NumericalHydraulics

Weinmann and Laurensdn, E.M. (1979) ApproximateFlood Rou-t— .g Methods : A Review, Journal of the Hydrau­lics Division, ASCE, Vol.105 Mo. HY12, pp 1521 - 1536.

Author Gibbons Francis John GeorgeName of thesis A Hydrodynamic Flood Routing Model For One Dimensional Flow. 1986

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